A Dynamical Approach to Non-Extensive Thermodynamics

A D YNAMICAL APPR O A CH TO NON-EXTENSIVE THERMOD YNAMICS AR TUR O. LOPES AND P A ULO V ARAND AS Abstract. W e dev elop a non-extensiv e thermo dynamic formalism for the one-sided shift on a ﬁnite alphab et, inspired by Tsallis’ generalization of Boltzmann en trop y in statistical ph ysics. W e in tro duce notions of q -en tropy , q -pressure and q -transfer op erators whic h extend the classical thermo dynamic formalism when q = 1. W e pro v e a Bo w en- t yp e relation linking the q -pressure with a (2 − q )-Ruelle transfer op erator and show that q -equilibrium states correspond to classical equilibrium states for a related potential. W e establish existence and uniqueness of q -equilibrium states for Lipschitz p oten tials, prov e diﬀeren tiability of the q -pressure, and obtain v ariational principles for b oth the q -pressure and a related asymptotic pressure. Finally , we study cohomological equations asso ciated with (2 − q )-transfer op erators and prov e diﬀerentiable dep endence of their solutions on the potential, yielding an alternative construction of eigenfunctions for classical Ruelle op erators. W e also prop ose an approac h to non-extensive thermo dynamics using non- additiv e formalisms. 1. Intr oduction 1.1. Non-extensiv e en trop y. En trop y of in v arian t measures is a fundamental concept used in dynamical systems to quantify the rate of information pro duction as a system ev olv es o v er time. Within the thermo dynamic formalism, this notion of metric entrop y pla ys a central role, allowing to relate the top ological complexity of the dynamical system with the largest p ossible complexity oﬀered by the inv arian t measures b y means of the classical v ariational principle for top ological pressure. In particular, if σ : Ω → Ω denotes the usual shift acting on the space Ω = { 1 , 2 , ..., d } N and A : Ω → R is a contin uous p oten tial then P ( A ) = sup n h ( µ ) + Z A dµ : µ ∈ M inv ( σ ) o (1.1) where P ( A ) denotes the topological pressure of A and h ( µ ) denotes the Kolmogoro v- Shannon en trop y of the σ -in v ariant probabilit y measure µ , computed through dynamically deﬁned partitions which are w eigh ted according to the Boltzman entrop y function H whic h, to each probability v ector p = ( p 1 , p 2 , . . . , p d ) asso ciates the v alue H ( p ) = d X i =1 − p i log( p i ) (1.2) 1 A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 2 (see e.g. [ 77 ] for the deﬁnitions and proof ). In a non-dynamical framework, the previous expression coincides with Shannon information, sometimes referred to as static entr opy . Sev eral scien tiﬁc pap ers consider concepts of entr opy which diﬀer from the Kolmogoro v- Shannon en tropy ( 1.2 ), whose emphasis is to provide a bias on rare even ts (see e.g. [ 4 , 15 , 18 , 42 , 66 , 69 , 75 ] and the discussion therein). In fact, in Ph ysics’ literature it is somewhat common to consider a parameterized concepts of en trop y , where for cer- tain parameters en tropy b ecomes non-additive even if dealing with independent systems (cf. [ 2 , p.75 equation (6)]). In order to elaborate further on that, let us recall some deﬁnitions considered in the classical literature on the non-extensive en tropy theory (in a non-dynamically framework). Giv en q > 0, the q -entr opy of the probability v ector p = ( p 1 , p 2 , ..., p n ), in tro duced b y Havrda and Charv at [ 35 ] and Tsallis [ 70 ], is deﬁned as H q ( p ) = 1 1 − q ( d X i =1 p q i − 1) = 1 1 − q d X i =1 p i ( p q − 1 i − 1) = d X i =1 p i log q  1 p i  ⩾ 0 , (1.3) and, for eac h q  = 1, the function R + ∋ u 7→ log q ( u ) = 1 1 − q ( u 1 − q − 1) (1.4) is called the q -lo g function . The case q = 1 which corresp onds to Kolmogoro v-Shannon en trop y . It is clear that if q  = 1 and and one w an ts to maximize 1 1 − q ( P i p q i − 1) among diﬀeren t probabilit y vectors p then there exists a bias which is not presen t whenev er q = 1. Indeed, if q < 1 then p q i > p i and the q -entrop y will enhance the relative imp ortance of rare ev ents, and if q > 1 one will get the opposite. W e refer the reader to [ 69 ] for a discussion. Moreo v er, for a ﬁxed probability vector p , the limit of H q ( p ) , as q → 1 is the classical Boltzman en trop y H ( p ). In what follo ws w e will emphasize some of the prop erties of the q -entrop y function and then establish a comparison with the classical thermo dynamic formalism. Throughout this paper it will b e a standard assumption that q > 0, in whic h case it makes sense to maximize en trop y (see Figure 1 ). In fact, for q > 0, the q -entrop y Figure 1. Graph of the function p 1 → H q ( p 1 , 1 − p 1 ) when q = 0 . 9 (left) and q = − 0 . 1 (right) A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 3 function p 7→ H q ( p ) deﬁned by ( 1.3 ) is conca v e as a function of the probabilit y vector p = ( p 1 , p 2 , .., p n ) b ecause d 2 dp 2 j 1 1 − q p q j = 1 1 − q q ( q − 1) p q − 2 j < 0 and the ﬁnite sum of concav e functions is concav e. Moreo v er, there is a crucial diﬀerence b et w een q -log functions for q = 1 and q  = 1, with striking impact on the corresp onding notions of extensive and non-extensiv e entropies, resp ectiv ely . Indeed, for every q  = 1 and a, b > 0, log q ( ab ) = log q ( a ) + log q ( b ) + (1 − q ) log q ( a ) log q ( b )  = log q ( a ) + log q ( b ) the joint indep enden t probabilit y v ector r s obtained from probability vectors r, s satisﬁes the non-extensive relation H q ( r s ) = H q ( r ) + H q ( s ) + (1 − q ) H q ( r ) H q ( s ) (1.5) and one reco v ers additivit y ab o v e if and only if q = 1 (cf. [ 1 ]), which corresp onds to the classical (extensiv e) Boltzman en trop y . W e refer the reader to Section 12 , to [ 78 ] or [ 73 , App endix, page 84] for for more details on q -exp onen tial, q -logarithmic functions and q -en tropies. In this general non-dynamical framework, given constan ts q > 0, β ∈ R and a p oten tial A : { 1 , 2 , ..., n } → R , one can deﬁne a notion of q -pr essur e b y the v ariational relation P q ( β A ) = sup p n H q ( p ) + β d X j =1 p j a j o , (1.6) where a j = A ( j ) and the supremum is tak en ov er all probabilit y vectors p = ( p 1 , p 2 , ..., p n ), in corresp ondence to what Umaro v and Tsallis [ 69 ] refer to as the ﬁrst choice of v ariational problem. A second alternative, where the pressure is deﬁned replacing P d j =1 p j a j b y P d j =1 p q j a j will not b e considered here. The latter ﬁnds a dual in the Statistical Mechanics literature, where in ( 1.6 ) the q - pressure is min us the Helmholtz free energy and the v alues a j corresp ond to the v alues of minus the Hamiltonian. Ho w ever, in con trast to the classical pressure function, the pressure function R ∋ β 7→ P q ( β A ) deﬁned by ( 1.6 ) is not globally conv ex nor conca v e and sev eral problems can o ccur when considering large ranges of β (cf. Remark 4.2 ). 1.2. Non-extensiv e thermo dynamic formalism for the shift. In this pap er we aim to develop a the dynamical non-extensive thermo dynamic formalism in parallel with the extensiv e (classical) setting, lo oking for p ossible matches and discrepancies with some of the well known results in the classical thermo dynamic formalism dev elop ed in the last decades (see e.g. [ 6 , 14 , 10 , 22 , 24 , 28 , 32 , 41 , 43 , 37 , 40 , 57 , 54 , 58 , 62 , 65 , 47 , 67 ] just to mention a few). Our work should b e considered as an initial attempt to explore and prop ose alternative answers to some questions that in our view are fundamen tal in the theory . F or that reason, w e in vestigate a dynamical v ersion of this class of problems for the one-sided shift on a ﬁnite alphab et, where the extensive thermo dynamic formalism A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 4 (whic h corresp onds to the classical Thermo dynamic F ormalism, using the Kolmogorov- Shannon en trop y) is extremely well understo o d. Although b oth theories hav e similar motiv ation, the non-extensive thermo dynamic formalism presents challenges and sev eral conceptual diﬀerences in resp ect to the classical (extensiv e) thermo dynamic formalism and non-additiv e thermo dynamic formalism which we will now discuss in detail. Consider the shift σ : Ω → Ω acting on the symbolic space Ω = { 1 , 2 , ..., d } N . W e denote b y M inv ( σ ) the set of σ -in v ariant probabilit y measures and b y G the set of classical (or extensiv e) equilibrium states asso ciated to Lipsc hitz contin uous potentials. These measures are all ergodic, are singular with respect to each other and parameterized b y the asso ciated Jacobian function J µ (cf. ( 2.3 ) and [ 41 ] for more details). Moreo v er, probabilit y measures in G ha ve nice ergo dic prop erties, as they are mixing, ha ve exp onen tial decay of correlations for H¨ older con tin uous observ ables and satisfy large deviation principles (see e.g. [ 10 , 50 ]). F urthermore, by Rohklin’s form ula, the Kolmogorov-Shannon en trop y h ( µ ) asso ciated to µ ∈ G is given by h ( µ ) = − Z log J µ dµ. (1.7) The extensive pressure P ( A ) of a Lipschitz contin uous p oten tial A : Ω → R satisﬁes the classical v ariational principles P ( A ) = sup n h ( µ ) + Z A dµ : µ ∈ M inv ( σ ) o = sup n h ( µ ) + Z A dµ : µ ∈ G o (1.8) (the second equalit y in ( 1.8 ) is due to the upper-semicontin uit y of the entrop y map and that an y in v arian t probability can b e weak ∗ appro ximated, and in en tropy , by a probability measure in G , cf. [ 50 ]). It is also well known that there exists a unique equilibrium state µ A ∈ G which maximizes the previous expression. Giv en a Lipschitz contin uous p oten tial A : Ω → R , Ruelle’s theorem relates sp ecial features of the equilibrium state µ A with leading eigenv alue and eigenfunction for the R uel le tr ansfer op er ator L A : C 0 (Ω , R ) → C 0 (Ω , R ), which is the b ounded linear op erator giv en by L A ( f ) = d X a =1 e A ( ax ) f ( ax ) (1.9) for ev ery contin uous function f : Ω → R , where ax ∈ Ω denotes the sequence in σ − 1 ( x ) starting with the symbol a . Namely , the equilibrium state µ A is such that µ A = h A ν A where L A h A = λ A h A , L ∗ A ν A = λ A ν A and λ A = e P ( A ) > 0 is the simple leading eigenv alue of L A (see e.g. [ 6 ]). W e pro ceed to extend the ab o v e concepts to the non-extensiv e framew ork. A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 5 Non-extensive thermo dynamic quantities. Let us no w deﬁne the concepts of q -en trop y of an in v arian t measure in G , q -pressure function and q -equilibrium states, in a wa y that one can recov er the classical framew ork as the limit of such quan tities as q → 1. Inspired b y Rokhlins formula we deﬁne, for each q > 0, q  = 1, the q -entr opy of a probability measure µ ∈ G as H q ( µ ) = Z log q  1 J  dµ = 1 1 − q Z  J q − 1 − 1  dµ. (1.10) W e extend the concept of q − en trop y for probability measures µ ∈ M inv ( σ ) and show that the q -en trop y function is concav e and upp er semi-contin uous (see Deﬁnition 3.3 , and Lemmas 3.4 and 3.6 ). In the extensive framew ork Boltzman en tropy is conca ve and the Kolmogorov-Shannon en trop y is an aﬃne function on the con vex set of σ -inv arian t probabilit y measures (see Theorem 8.1 page 183 in [ 77 ]). How ev er, the dynamical q - en trop y is concav e when 0 < q ⩽ 1 (cf. Example 12.1 ). In view of the second equality in ( 1.8 ), giv en a contin uous p oten tial A : Ω → R w e deﬁned the dynamical q -pr essur e function of A by the v ariational relation P q ( A ) = sup n H q ( µ ) + Z A dµ : µ ∈ G o , (1.11) and we will sa y that µ q is a q -e quilibrium state with resp ect to the potential A is an in v ariant Gibbs measure attaining the suprem um in ( 1.11 ). By deﬁnition, the previ- ous supremum is taken o v er the space of extensive Gibbs equilibrium states G , hence if non-extensiv e equilibrium states exist then these are equilibrium states for the classical thermo dynamic formalism. In order to dev elop a sp ectral approac h for the non-extensive thermo dynamic formalism one needs to consider suitable transfer op erators. Given q > 0, with q  = 1, the inv erse of log q is the q -exp function deﬁned b y u 7→ e u q = exp q ( u ) = (1 + (1 − q ) u ) 1 1 − q . (1.12) This suggests to consider the family of transfer op erators L A,q : C 0 (Ω , R ) → C 0 (Ω , R ) as L A,q ( f ) = d X a =1 e A ( ax ) q f ( ax ) = d X a =1 [1 + (1 − q ) A ( ax )] 1 1 − q f ( ax ) , f ∈ C 0 (Ω , R ) , (1.13) whenev er the latter is well deﬁned. A t this p oin t there are ma jor technical and conceptual diﬀerences in resp ect to the extensiv e transfer op erators, at b oth conceptual and technical viewp oin ts. F rom the tec hnical viewp oint, the q -exp function b ehav es in a quite intricate w ay: (i) if q > 1 then exp q ( u ) is p ositiv e if u < 1 q − 1 ; (ii) if 0 < q < 1, the v alue exp q ( u ) is p ositiv e whenever u > 1 q − 1 and is complex otherwise (up to q = 1 / 2, where is alwa ys a non-negativ e real n um b er). Moreo v er, for q > 0, the q -exp function is con vex . The main conceptual A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 6 diﬀerences in resp ect to the extensive thermo dynamic formalism are describ ed in the next subsection. 1.3. Non-extensiv e op erators: duality and non-additiv e thermo dynamic for- malism. Let us no w describ e the dynamical framew ork for non-extensiv e thermo dynamic formalism, developed in the pap er. The ﬁrst key observ ation is that there exists a non- standard relation on the q -parameter in terv al I = (0 , 2) at q = 1: the q -equilibrium states and the q -pressure function relates to the (2 − q )-Ruelle op erator. More precisely , suc h a relation q − pressure function P q ( · ) ↭ transfer op erator L A, 2 − q ( · ) is formalized in Theorem A , which oﬀers a Bow en-t ype form ula, where w e prov e that solutions of a log-functional equation in v olving the (2 − q )-transfer operators are related to a zero of a Bo w en-t yp e equation in v olving the q -pressure function. In this wa y , among the non-extensiv e q -equilibrium states associated to a certain Lipsc hitz con tin uous p oten tial A there exist extensiv e (classical) equilibrium states for a related Lipsc hitz con tin uous (cf. Theorem A for the precise formulation). The statement of Theorem A is far from establishing a dictionary b etw een extensiv e and non-extensive thermo dynamic formalism. A fact that reinforces the latter is that the q -pressure function β → P q ( β A ) is neither conv ex nor concav e on β for large ranges of β (see Remark 4.2 ), whic h creates technical problems for the use of the classical Legendre transform. In this wa y , setting the duality of MaxEnt method and pressure in the non- extensiv e dynamical via the classical Legendre transform formalism seems not to ﬁnd parallel in the non-extensive framew ork (compare the deriv ativ es of the pressure functions in [ 62 , Prop osition 4.10], Example 8.1 and ( 9.2 ) in case q = 1 / 2). F urthermore, the lack of con v exity of the q -pressure function seems to contribute to a muc h ric her structure on the space of q -equilibrium states and there are examples where the log-functional equation has non-unique solutions (see Section 8 ). Another striking diﬀerence to the extensive thermo dynamic formalism can b e observed at the level of q -transfer op erators, for each non-negativ e q  = 1. Indeed, while the usual exp onen tial function exp : ( R , +) → ( R + , × ) is a group homomorphism and the leading eigenfunction of the classical transfer op erators can b e obtained b y normalized iterates L n A, 1 (1), one has that e a + b q  = e a q e b q for every q > 0, q  = 1, and ev ery non-zero a, b ∈ R . In this wa y , the weigh ts app earing in the iterates L n A,q (1) are muc h more intricate than Birkhoﬀ sums, which are classical to Boltzman-equilibrium statistics. In order to o ver- come this fact w e introduce a sequence ( L n ) n ⩾ 1 of transfer op erators ( 2.9 ) adapted to the shift and a family Φ = ( φ n ) n ⩾ 1 of Lipschitz con tinuous p oten tials φ n : Ω → R giv en by ( 2.10 ), whic h falls in the realm of non-additiv e thermo dynamic formalism. The sequence Φ = ( φ n ) n ⩾ 1 has extremely mild additivity , i.e. it is just asymptotically sub- additiv e. Nev ertheless one can show that the q -asymptotic pressure P q ( A ), deﬁned to represen t the exp onen tial gro wth rate of the norms of transfer op erators L n , satisﬁes a A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 7 v ariational principle in volving the Kolmogoro v-Shannon metric entrop y and that equilib- rium states alw ays exist (see Theorem B and Lemma 6.2 for the precise statements). In Theorem C we prov e that the solutions of the functional equation concerning the (2 − q )-Ruelle transfer op erators v ary diﬀerentiably with the p oten tial on a neighborho o d of the original p oten tial A , pro vided that this is normalizable. This result can also be used to pro vide an alternativ e argument for the existence of eigenfunctions for transfer op erators, using the implicit function theorem. Finally , there are several natural notions of equilibrium states app earing motiv ated by the non-extensiv e thermo dynamic formalism, and it is of huge interest to understand their in terpla y . A general relation b etw een t wo of these notions, namely q -equilibrium states and q -asymptotic equilibrium states, still seems out of reach due to the muc h diﬀerent nature of the non-extensiv e ob jects (cf. Remark 2.3 ). Nev ertheless, as Theorem A establishes a bridge b et w een non-extensiv e q -equilibrium states of a Lipsc hitz con tin uous p oten tial A and extensive equilibrium states for a modiﬁed Lipsc hitz con tin uous p otential, it is natural to lo ok for the dep endence of these ob jects on the p oten tial A , and to understand its p ossible applications. 1.4. Organization of the pap er. F or the readers’ con venience let us give a brief des- cription on the organization of this pap er. The main results of this pap er are stated in Section 2 : Theorem A oﬀers a dualit y b et w een the q -pressure function and the (2 − q )-Ruelle op erator, Theorem B establishes a v ariational principle for the non-extensiv e pressure function, while Theorem C describ es solutions of a certain cohomological equation. Our main fo cus in the presen t pap er is the study of the concept of q -entrop y for prob- abilit y measures µ ∈ G . In Section 3 w e describe prop erties of q -entrop y functions H q ( · ) for Gibbs and Bernoulli measures. Later, in Section 4 we consider a dynamical point of view for q -pressure under the non-extensiv e framework. First we analyze the case where the probabilit y p is of the form p = ( p 1 , p 2 , ..., p n ) and w e exhibit the maximal q -pressure probabilit y for a given p otential (cf. Section 4.1 ). This corresp onds to the case in whic h the dynamical system do es not interv ene and it is in consonance with most results in non- extensiv e Statistical Mechanics (see e.g. [ 26 , p. L71]), as our deﬁnitions are asso ciated to the so called ﬁrst c hoice of MaxEnt metho d as describ ed in [ 26 , 70 , 74 ]. Our main ob jectiv e in Section 4.1 is to allow the reader to understand the nature of the results that will b e extended in later sections, and that will contemplate the dynamical viewp oint. In Subsection 4.2 considers the non-extensiv e p oint of view for the q -pressure in a dynamical setting. In Section 5 w e pro ve Theorem A . In order to do so, in Section 5.3 w e consider metho ds related to the problem of solving the (2 − q )-Ruelle op erator equation. In Subsections 5.3.1 and 5.3.2 , for 0 < q < 1, w e exhibit p otentials A for whic h w e can ﬁnd solutions for A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 8 the (2 − q )-Ruelle op erator equation ( 2.5 ), as solutions for the classical Ruelle op erator equation with resp ect to some mo diﬁed p oten tials. In Section 6 we relate the q -asymptotic pressure function with results from classical non- additiv e thermo dynamic formalism. First, in Subsection 6.1 w e describ e the asymptotic sub-additiv e prop ert y of the family of p oten tials ( φ n ) n ⩾ 1 , in the case 0 < q < 1. This will b e crucial in the pro of of Theorem B , which app ears in Subsection 6.2 . On Section 7 w e presen t a pro of of the existence of solutions for q -Ruelle Theorem equa- tion using a v ersion of the implicit function theorem (see [ 45 , 5 ]), and prov e Theorem C . In Section 8 is devoted to sp eciﬁc classes of examples on which the non-extensive ob jects can be computed. In the case of lo cally constan t p otentials A : { 1 , 2 } N → R whic h dep end only on the ﬁrst t w o co ordinates, that is A ( x ) = A ( x 1 , x 2 ), we will sho w that the dynamical q -equilibrium state is a Marko v probabilit y . Using a diﬀerent technique, Example 8.5 we presen t explicit solutions for the pressure problem in a non-extensiv e setting b y exploring a relation b et w een these and the q -Ruelle op erator acting on Lipschitz contin uous p otentials, hence obtaining a non-extensive v ersion of the Ruelle’s Theorem. The remainder of the paper consists of four app endices. App endix A in Section 9 describ es an explicit expression for the deriv ative of the pressure in the case q = 1 / 2. App endix B in Section 10 describ es the point of view of dynamical partitions for the non-extensiv e case (someho w related to [ 60 ]). In App endix C Section 11 w e brieﬂy relate the results of the non-extensiv e en trop y describ ed in our text with Ren yi entr opy . Finally , the App endix D (Section 12 ) contains a self-contained description of several general and useful prop erties for the q -log and q -exp functions; some of them are used throughout the pap er. 2. Main resul ts 2.1. Setting. Let σ : Ω → Ω denote the one-sided shift on Ω = { 1 , 2 , ..., d } N , endow ed with the usual distance (whic h w e denote b y dist), whic h mak es it diameter one. Let C 0 (Ω , R ) b e the Banach space of all contin uous functions on Ω endo w ed with the C 0 - top ology and let Lip(Ω) ⊂ C 0 (Ω , R ) b e the subspace of Lipsc hitz contin uous functions endo w ed with the norm ∥ f ∥ Lip = ∥ f ∥ C 0 + | f | Lip where | f | Lip = sup x  = y | f ( x ) − f ( y ) | dist( x, y ) (2.1) Giv en a Lipschitz contin uous function A : Ω → R the R uel le tr ansfer op er ator asso ciated to the p oten tial A is the linear op erator L A giv en by ( 1.9 ). Its dual, denoted by L ∗ A , acts on the space M 1 (Ω) of probabilit y measures on Ω b y the dualit y relation Z f d ( L ∗ A ( µ 1 )) = Z L ( f ) dµ 1 ∀ f ∈ C 0 (Ω , R ) . A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 9 It is clear from the deﬁnition that L A preserv es the space Lip(Ω) of Lipsc hitz con tinuous p on ten tials on Ω. In case there exists c ∈ R and a contin uous function φ : Ω → R + so that the Ruel le op er ator e quation L A  λ − 1 φ ( φ ◦ σ )  ( x ) = d X a =1 e A ( ax )+log( φ ( ax )) − log( φ )( x ) − c = 1 for every x ∈ Ω (2.2) holds then w e sa y that A is a normalize d p otential , that φ is the eigenfunction and that the constan t λ = e c > 0 is the the eigenvalue asso ciated to L A . F or short we will say that the pair ( φ, c ) solv es the (extensiv e) Ruelle op erator equation. Ruelle’s theorem ensures that for any Lipsc hitz con tin uous potential A : Ω → R there exist c ∈ R and contin uous p ositiv e Lipschitz con tin uous function φ : Ω → R satisfying ( 2.2 ), and that the top ological pressure P ( A ) of the shift σ with resp ect to the potential A coincides with log λ (see e.g. [ 6 , 62 ]). A Jac obian J : Ω → R is a positive Lipschitz contin uous function such that L log J (1)( x ) = d X a =1 J ( ax ) = 1 for every x ∈ Ω. (2.3) F or eac h Jacobian J there exists a unique probability measure µ = µ log J , fully supp orted, suc h that L ∗ log J ( µ ) = µ , to which we will refer as the (extensiv e) e quilibrium state of the p oten tial log J (these are often called Gibbs me asur es cf. [ 10 ]). The space G = n µ log J : J is a Jacobian o (2.4) of all Gibbs measures asso ciated to the extensive thermo dynamic formalism is an inﬁnite dimensional manifold (cf. [ 55 ]). As there exists a bijective relation b et ween the space of Jacobians and the asso ciated equilibrium states, the elemen ts in G can be either param- eterized by their elemen ts µ or by their Jacobians J (w e refer the reader to [ 41 , 55 ] for more details). Our main goal is to determine whether there exist q -equilibrium states - re- call these are probability measures on G attaining the suprem um in ( 1.11 ) - and to build p ossible bridges b et w een the classical (extensiv e) equilibrium states and q -equilibrium states. 2.2. Statemen ts. The starting p oin t for our study of the non-extensive thermo dynamic formalism is the following dualit y b etw een the q -pressure function and the (2 − q )-Ruelle op erator equation. In this wa y , the next theorem provides not only suﬃcient conditions for the solution of a non-extensive Bow en t yp e equation, as it ensures that the q -equilibrium state for a given Lisp c hitz p otential A is a classical equilibrium for a pressure problem for another p oten tial within the extensive thermo dynamic formalism. A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 10 Theorem A. L et q > 0 and A : Ω → R b e a Lipschitz c ontinuous p otential. If ther e exists a c onstant c ∈ R and a c ontinuous function φ : Ω → R so that d X a =1 e A ( ax )+ φ ( ax ) − φ ( x ) − c 2 − q = 1 for every x ∈ Ω (2.5) and that al l summands ab ove ar e strictly p ositive then P q ( A ) = c . Mor e over, the fol lowing pr op erties hold: (1) ther e exists a q -e quilibrium state asso ciate d to A that c oincides with the e quilibrium state for the p otential log J , wher e J ( x ) = e A ( x )+ φ ( x ) − φ ( σ ( x )) − c 2 − q ; (2) P q  − log q  1 J  = 0 , (2.6) wher e J is the classic al Jac obian describ e d by − log q  1 J  = − 1 + (1 + ( A + φ − ( φ ◦ σ ) − c ) ( q − 1)) − 1 q − 1 . (2.7) R emark 2.1 . Theorem A mak es explicit a symmetry b etw een the parameter q of the q -pressure and the parameter ˜ q = 2 − q app earing in the ˜ q -Ruelle transfer op erator equation ( 2.5 ). In the extensiv e framework (corresp onding to q = 1) we reco v er that the pressure function can b e obtained through the leading eigenv alue of the classical transfer op erator. W e will refer to a solution φ of ( 2.5 ) as the (2 − q ) -non-extensive lo g- eigenfunction (eigenfunction for short) and to the constan t c as the (2 − q ) -non-extensive lo g-eigenvalue (eigen v alue for short). The existence of a contin uous function φ and c ∈ R solving ( 2.5 ) is a non-extensiv e v ersion of the Ruelle op erator theorem [ 62 , Theorem 2.2]. R emark 2.2 . The pair of solutions φ, c in Theorem A ne e d not to b e unique (see Example 8.4 ). Moreo v er, in Example 8.5 we exhibit a Lip chitz contin uous p oten tial A where there exist a pair of solution φ, c for ( 2.5 ), but one of the summands can take the v alue zero for some x ∈ Ω. In this wa y , in the non-extensiv e setting one should not exp ect a full extension of the classical Ruelle Theorem. It is worth mentioning that whenev er one writes an expression like ( 2.5 ) w e are implicitly assuming that all v alues are well deﬁned and, in most cases, we will assume that all summands are strictly p ositive for ev ery x . Let us discuss a bit further the class of p oten tials that one considers. Giv en 1 < q ⩽ 2 consider the op en subset (in the Lipschitz top ology) F q = n A ∈ Lip(Ω) : A ( x ) > 1 1 − q , ∀ x ∈ Ω o . A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 11 This op en set v aries contin uously with q , in the Hausdorﬀ distance. A t a ﬁrst sigh t Theorem A could suggest to consider, for each A ∈ F q , the q -transfer op erator L A,q ( f ) = d X a =1 e A ( ax ) q f ( ax ) , f ∈ C 0 (Ω , R ) Ho w ever, in virtue of the prop erties of q -exp onen tials (cf. Section 12 ) their iterates L n A,q ( f )( x ) = X σ n ( y )= x h n − 1 Y j =0 e A ( σ j ( y )) q i f ( y ) seem not suitable to study the non-extensiv e thermo dynamic formalism. F or that reason w e will consider the family of transfer op erators ( L n ) n ∈ N , deﬁned b y L n ( f )( x ) = X σ n ( y )= x e A ( y )+ A ( σ ( y ))+ ... + A ( σ n − 1 ( y )) q f ( y ) (2.8) for ev ery f ∈ C 0 (Ω , R ) and x ∈ Ω, whic h can b e written as L n ( f )( x ) = X σ n ( y )= x e φ n ( y ) f ( y ) (2.9) where φ n ( y ) = 1 1 − q log 1 + (1 − q ) n − 1 X j =0 A ( σ j ( y )) ! , n ∈ N . (2.10) The family of p otentials ( φ n ) n ⩾ 1 is asymptotically sub-additive (cf. Lemma 6.2 ). Ho w- ev er, this family of p otentials is not almost additiv e and do es not seem to satisfy an y of the suﬃcient conditions that allo w to study the non-additiv e thermo dynamic formal- ism developed in previous w orks (see [ 11 , 17 , 30 , 76 ] and references therein), which do es not allow us to use of sp ectral theory to study non-extensiv e thermodynamic formalism. Nev ertheless we pro ve the following v ariational principle. Theorem B. Given 0 < q < 1 , a Lip chitz c ontinuous p otential A : Ω → R the limit P q ( A ) = lim n →∞ 1 n log L n (1)( x 0 ) (2.11) exists and is indep endent ot the p oint x 0 . Mor e over, P q ( A ) = max n h ( ν ) + lim n →∞ 1 n Z φ n dν : ν ∈ M inv ( σ ) o , (2.12) wher e h ( ν ) is the extensive Kolmo gor ov-Shannon entr opy of ν . The proof of Theorem B will b e giv en in Section 6 . The limit P q ( A ) in ( 2.11 ) will b e called the q -asymptotic pr essur e of the p otential A : Ω → R and the probabilit y measures attaining the maxim um will b e referred to as q -asymptotic e quilibrium states . A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 12 R emark 2.3 . The v ariational principle in ( 2.12 ) should be compared with the v ariational deﬁnition of the q -pressure P q ( A ) in ( 1.11 ). The exp onen tial gro wth rate of the norm of the op erators ( L n ) n ∈ N is related to sum of extensive en tropies with the av erage of a sub-additiv e family of p otentials, whereas the q -pressure considers non-extensiv e entropies but considers the usual integral ov er the p otential. R emark 2.4 . A priori there is no relation b etw een q -asymptotic equilibrium states and the (non-extensive) q -equilibrium states, as these inv olv e diﬀerent measure theoretical en tropies. The ab o v e expression connects the non-extensive setting with the sub-additive setting (recall ( 1.3 )). Theorem C. L et 0 < q ⩽ 1 , let ˜ A : Ω → R b e a normalize d Lipschitz c ontinuous p otential and let ν ˜ A b e the pr ob ability me asur e such that L ∗ ˜ A ν ˜ A = ν ˜ A . Ther e exists an op en neighb orho o d U ⊂ Lip (Ω) of ˜ A and a diﬀer entiable map U ∋ A 7→ ( φ A , c A ) ∈ Lip (Ω) × R such that d X a =1 e A ( ax )+ φ A ( ax ) − φ A ( x ) − c A q = 1 for every x ∈ Ω (2.13) and R φ A dν ˜ A = 0 . Mor e over, lim A → ˜ A ( φ A , c A ) = (0 , 0) ∈ Lip (Ω) × R . The previous result will b e prov ed in Section 7 , stated as Theorem 7.1 . One can ask ab out natural condition under which for a Lipsc hitz con tinuous p oten tial there exist φ and c such that ( 2.13 ) holds. W e conjecture that giv en q > 0, in general there exists a p ositiv e answer to the question for p otentials A on some op en set of p oten tials. In the sp ecial case that Ω = { 1 , 2 } N , and A dep ends on t w o coordinates, we obtain explicit solutions for a non trivial class of examples (see Example 8.5 in Section 8 and Subsection 5.3.1 ). R emark 2.5 . W e p oin t out that when considering p oten tials whic h are merely con tin uous in the extensiv e thermo dynamic formalism then phase transitions, slow deca y of corre- lations and existence of non-ergodic equilibrium states may app ear (see e.g. [ 21 , 22 , 33 , 46 , 48 , 51 , 47 ] and references therein). In particular there exist con tin uous p oten tials A : Ω → R for whic h there is no eigenfunction solution for the Ruelle operator equation ( 2.2 ) (see [ 41 ]). In this pap er we will alw a ys consider Lipsc hitz contin uous p oten tials and p ostp one the study of the non-extensive thermo dynamic formalism for less regular p oten tials to a future work. R emark 2.6 . The extensiv e equilibrium state ˆ µ for the tw o-sided shift and a Lipschitz con tin uous p otential ˆ A : { 1 , 2 , ..., d } Z → R can b e analyzed by showing that it is coho- mologous to a Lipsc hitz contin uous p otential A : { 1 , 2 , ..., d } N → R which dep ends only of p ositiv e co ordinates (see Prop osition 1.2 in [ 62 ] or App endix 7.1 in [ 59 ]). A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 13 3. q -entropies In this subsection we will study some basic prop erties of q -en tropy functions, namely that this is a conca v e and upp er-semicon tin uous function on the probabilit y measures. Throughout the section let C + denote the space of con tin uous p ositiv e functions on { 1 , 2 , .., d } N . 3.1. Gibbs measures. Giv en 0 < q < 1, recall the q -en tropy of a probability measure µ ∈ G is H q ( µ ) = Z 1 1 − q ( J ( x ) q − 1 − 1) dµ ( x ) = Z log q ( 1 J ( x ) ) dµ ( x ) where J = J µ is the Jacobian of µ ∈ G , hence it is alw ays non-negativ e (see Section 12 for the prop erties of the q -exp onential function). Moreov er, as − log x > 1 1 − q ( x q − 1 − 1) for ev ery 0 < x < 1 and 0 < q < 1, one has that h ( µ ) ⩾ H q ( µ ) for ev ery µ ∈ G and 0 < q < 1. (3.1) Lemma 3.1. The function G ∋ µ → H q ( µ ) is diﬀer entiable. Pr o of. By [ 55 ], the space G is an analytic Banac h manifold. Therefore, for a ﬁxed q > 0, using that log q is diﬀeren tiable on its domain and that the function µ → J µ is diﬀeren tiable (see e.g. [ 41 ]). Hence we conclude that G ∋ µ → H q ( µ ) is diﬀeren tiable as well. □ W e pro ceed to prov e a v ariational c haracterization for the q -en trop y of Gibbs measures. Lemma 3.2. Fix an H¨ older c ontinuous normalize d p otential B = log J : { 1 , 2 , .., d } N → R and let µ log J b e the e quilibrium state with Jac obian J . Then, for 0 < q < 1 , H q ( µ ) = inf u ∈C +  Z log q  P d a =1 u ( a x ) u ( x )  dµ ( x )  . (3.2) Pr o of. By deﬁnition in ( 1.10 ) one kno ws that H q ( µ ) = R log q ( J − 1 ) dµ. T aking ˜ u ( x ) = e log J ( x ) ∈ C + , one can write the right-hand side ab ov e as log q  P d a =1 ˜ u ( a x ) ˜ u ( x )  = log q  P d a =1 J ( a x ) e log J ( x )  = log q ( J − 1 ) , and so, b y integration, Z log q  P d a =1 ˜ u ( a x ) ˜ u ( x )  dµ ( x ) = H q ( µ ) . No w, giv en a general u ∈ C + it can alwa ys b e written as u = uJ, where u is p ositiv e. W e claim that Z log q  P d a =1 u ( a x ) u ( x )  dµ ( x ) ⩾ H q ( µ ) . A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 14 Using that log( y ) ⩽ log q ( y ) for eac h 0 < q < 1, the concavit y of x → log ( x ) and Jensen’s inequalit y we deduce that Z log q  P d a =1 u ( a x ) u ( x )  dµ ( x ) = Z log q  P d a =1 u ( a x ) J ( ax ) u ( x ) J ( x )  dµ ( x ) ⩾ Z log  d X a =1 u ( a x ) J ( ax ) u ( x ) − 1 J ( x ) − 1  dµ ( x ) = Z log  d X a =1 [ u ( a x ) u ( x ) − 1 J ( x ) − 1 ] J ( ax )  dµ ( x ) ⩾ Z d X a =1 log( u ( a x ) u ( x ) − 1 J ( x ) − 1 ) J ( ax ) dµ ( x ) . This, together with the σ -inv ariance of µ and ( 3.1 ) yields that the right-hand side term ab o v e can b e written as Z log( u ( x ) u ( σ ( x )) − 1 J ( σ ( x )) − 1 ) dµ ( x ) = Z log( J ( σ ( x )) − 1 ) dµ ( x ) = Z log( J ( x ) − 1 ) dµ ( x ) = h ( µ ) ⩾ H q ( µ ) . This pro ves the lemma. □ 3.2. In v arian t measures. In this subsection we extend the concept of q -en trop y for arbitrary probabilit y measures in M inv ( σ ) . In fact, Lemma 3.2 suggests the follo wing deﬁnition. Deﬁnition 3.3 . Giv en 0 < q ⩽ 1 and a σ -inv arian t probability measure µ ∈ M inv ( σ ), the q − entr opy of µ is deﬁned as H q ( µ ) = inf u ∈C +  Z log q  P d a =1 u ( a x ) u ( x )  dµ ( x )  . (3.3) W e note that, in opp osition to the case of Gibbs measures, for an arbitrary inv arian t measure the inﬁm um in ( 3.3 ) may not b e necessarily attained b y some function in C + . W e pro ceed to study the concavit y of the q -entrop y map µ 7→ H q ( µ ). Conca vit y will not follo ws from a naive approac h using Gibbs measures as, while for each λ ∈ (0 , 1) and µ 1 , µ 2 ∈ G , the inv arian t probability measure λµ 1 + (1 − λ ) µ 2 do es not b elong to G , ev en though it can b e weak ∗ accum ulated by Gibbs measures (see [ 50 ] or [ 41 , Corollary 7.14]). In this wa y we will use the v ariational formulation for q -en trop y to pro ve that it is actually a concav e function. Lemma 3.4. Fix 0 < q < 1 . The q -entr opy map M inv ( σ ) ∋ µ 7→ H q ( µ ) is c onc ave. A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 15 Pr o of. Fix 0 < q < 1. Giv en λ ∈ [0 , 1] and µ 1 , µ 2 ∈ M inv ( σ ), consider the probability µ λ = λµ 1 + (1 − λ ) µ 2 . By deﬁnition, given ε > 0 there exists u ∈ C + suc h that H q ( µ λ ) ⩾ Z log q  P d a =1 u ( a x ) u ( x )  dµ λ − ε = Z log q  P d a =1 u ( a x ) u ( x )  d [ λµ 1 + (1 − λ ) µ 2 ] − ε = λ Z log q  P d a =1 u ( a x ) u ( x )  d µ 1 + (1 − λ ) Z log q  P d a =1 u ( a x ) u ( x )  dµ 2 ( x ) − ε. No w, as q -entrop y of eac h µ i is deﬁned b y an inﬁm um ov er C + of the expression in ( 3.3 ) one concludes that H q ( µ λ ) ⩾ λH q ( µ 1 ) + (1 − λ ) H q ( µ 2 ) − ε. As ε > 0 is arbitrary , this pro v es the concavit y of the q -en trop y map, as desired. □ R emark 3.5 . The Kolmogorov-Shannon entrop y map M inv ( σ ) ∋ µ 7→ h ( µ ) is w ell known to b e aﬃne. Simulations suggest that the q − entrop y is not aﬃne when 0 < q < 1 . Lemma 3.6. Fix 0 < q < 1 . The q -entr opy H q ( µ ) is an upp er semi-c ontinuous function of µ on M inv ( σ ) . Pr o of. Fix µ ∈ M inv ( σ ) and tak e a sequence ( µ n ) n ⩾ 1 in M inv ( σ ) conv erging to µ in the w eak ∗ top ology . Assume, by contradiction, there exists ε > 0, such that lim sup n →∞ H q ( µ n ) > H q ( µ ) + ε. Up to extracting a subsequence of the original sequence w e will assume that the limit in the left-hand side do es exist. Then there exists u ∈ C + and N ⩾ 1, such that H q ( µ n ) > H q ( µ ) + ε ⩾ Z log q  P d a =1 u ( a x ) u ( x )  dµ ( x ) (3.4) for ev ery n ⩾ N . By weak ∗ con v ergence, as u is con tinuous and strictly p ositiv e one gets lim n →∞ Z log q  P d a =1 u ( a x ) u ( x )  dµ n ( x ) = Z log q  P d a =1 u ( a x ) u ( x )  dµ ( x ) whic h, together with ( 3.4 ), shows that H q ( µ n ) > Z log q  P d a =1 u ( a x ) u ( x )  dµ n ( x ) , whic h is a contradiction with ( 3.3 ). □ A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 16 The next result, inspired b y [ 77 , Theorem 9.12] oﬀers a dual v ariational principle for the q -en tropy , sho wing that is coincides with the Legendre-F enc hel transform of the q -pressure function. More precisely , deﬁne ˆ P q ( g ) = sup n H q ( µ ) + Z g dµ : µ ∈ M inv ( σ ) o . (3.5) The next result shows that the top ological pressure ˆ P q determines the q -entrop y H q map. Prop osition 3.7. Supp ose µ ∈ M inv ( σ ) . Then, H q ( µ ) = inf g ∈ C (Ω , R ) n ˆ P q ( g ) − Z g dµ o . Pr o of. Fix µ 0 ∈ M inv ( σ ) and g ∈ C (Ω , R ). By ( 3.5 ), ˆ P q ( g ) − R g dµ 0 ⩾ H q ( µ 0 ) . By arbitrariness of g this prov es that inf g ∈ C (Ω , R ) n ˆ P q ( g ) − Z g dµ 0 o ⩾ H q ( µ 0 ) . (3.6) W e pro ceed to prov e the conv erse. T ake b > H q ( µ 0 ) . T ake C = { ( µ, t ) ∈ M inv ( σ ) × R : 0 ⩽ t ⩽ H q ( µ ) } . Note that for any µ ∈ M inv ( σ ) there exists t ⩾ 0 such that ( µ, t ) ∈ C . As the entrop y map H q is conca v e (see Lemma 3.4 ), we get that C is a compact conv ex set: if ( µ, t ) , ( ν , s ) ∈ C and λ ∈ [0 , 1] then λt + (1 − λ ) s ⩽ λH q ( µ ) + (1 − λ ) H q ( ν ) ⩽ H q ( λµ + (1 − λ ) ν ) , which pro v es that ( λµ + (1 − λ ) ν , λt + (1 − λ ) s ) ∈ C . F rom the upp er semicontin uit y of H q (recall Lemma 3.6 ), one concludes that ( µ 0 , b ) / ∈ C . As M inv ( σ ) ⊂ M ( X ) ≃ C ( X ) ∗ , by the geometric Hahn-Banac h separation theorem (cf. [ 29 , p. 417]) there exists a linear functional L : C ( X ) ∗ × R → R and α ∈ R so that L ( µ, t ) ⩽ α < L ( µ 0 , b ) for ev ery ( µ, t ) ∈ C . By Riesz’s represen tation theorem, there exists a con tinuous function v : Ω → R such that Z v dµ + αt < Z v dµ 0 + α b, for all ( µ, t ) ∈ C . T aking ( µ 0 , H q ( µ 0 )) ∈ C in the previous expression w e get that αH q ( µ 0 ) < αb , which shows that α > 0 . Therefore, H q ( µ ) + 1 α Z v dµ < b + 1 α Z v dµ 0 for an y µ ∈ M inv ( σ ) and so, taking the supremum ov er all inv arian t measures and recalling ( 3.5 ), ˆ P q  v α  ⩽ b + 1 α Z v dµ 0 . Th us b ⩾ ˆ P q  v α  − 1 α Z v dµ 0 ⩾ inf n ˆ P q ( g ) − Z g dµ 0 : g ∈ C (Ω , R ) o . A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 17 Since b > H q ( µ 0 ) is arbitrary we conclude that H q ( µ 0 ) ⩾ inf n ˆ P q ( g ) − Z g dµ 0 : g ∈ C (Ω , R ) o . This ﬁnishes the pro of of the prop osition. □ The next result shows that, even though for eac h 0 < q < 1 the q -entrop y maps lac ks the symmetry of the extensiv e en trop y map, all of these attain its maximal v alue at the Bernoulli measure with equal weigh ts. More precisely: Lemma 3.8. Fix 0 < q < 1 . The function G ∋ µ 7→ H q ( µ ) has a unique maximum, attaine d at the Bernoul l li me asur e µ 0 with e qual weights 1 /d , and H q ( µ 0 ) = log q ( d ) . Pr o of. Note ﬁrst that, b y Lagrange multipliers, it is not hard to chec k that the supremum of the function p = ( p 1 , p 2 , ..., p d ) → Φ( p ) = P d j =1 p q j − 1 is the v alue d 1 − q − 1 , and that it is only attained at p 0 = (1 /d, 1 /d, ..., 1 /d ). Hence, if J denotes the Jacobian of a probability measure µ ∈ G then, by ( 1.10 ), σ -inv ariance of µ and P d a =1 J ( ax ) = 1, H q ( µ ) = Z 1 1 − q ( J ( x ) q − 1 − 1) dµ ( x ) = 1 1 − q Z d X a =1 J ( ax )( J ( ax ) q − 1 − 1) dµ ( x ) = 1 1 − q Z ( d X a =1 J ( ax ) q − 1) dµ ( x ) . No w, as ( J ( ix )) 1 ⩽ i ⩽ d is a probability vector one has that Φ(( J ( ix )) 1 ⩽ i ⩽ d ) ⩽ d 1 − q − 1 for ev ery x ∈ Ω. In consequence, H q ( µ ) = 1 1 − q Z ( d X a =1 J ( ax ) q − 1) dµ ( x ) ⩽ 1 1 − q ( d 1 − q − 1) = H q ( µ 0 ) = log q ( d ) , whic h prov es the lemma. □ 3.3. Dynamical relativ e q -en trop y. Let us ﬁnish this section by introducing the con- cept of relativ e q -entrop y for Gibbs measures. Supp ose µ i ∈ G has Lipsc hitz contin uous Jacobian J i , i = 1 , 2. The r elative q -entr opy (also called q -KL diver genc e ) is the v alue H q ( µ 1 , µ 2 ) = Z log q  1 J 2  dµ 1 − Z log q  1 J 1  dµ 1 . (3.7) This function H q ( µ 1 , µ 2 ) is analytic on the pair ( µ 1 , µ 2 ) in the Banach manifold of Lipsc hitz equilibrium states G (cf. [ 55 ]). Moreov er, H q ( µ 1 , µ 2 ) is non-negativ e: using ( 11.3 ) we A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 18 obtain H q ( µ 1 , µ 2 ) = Z log q ( 1 J 2 ( x ) ) dµ 1 ( x ) − Z log q ( 1 ˜ J ( x ) ) dµ 1 ( x ) = Z d X a =1 J 1 ( ax ) h log q ( 1 J 2 ( ax ) ) − log q ( 1 J 1 ( ax ) ) i dµ 1 ( x ) ⩾ 0 . (3.8) 4. The q -pressure function 4.1. The non-dynamical q -pressure function. In this subsection we consider the q - pressure in a setting where there is no underlying dynamical system, the initial setting considered in [ 27 , 70 ]. Althouth the results in this subsection are not strictly necessary for reading the rest of the pap er, these oﬀer a motiv ation and insigh ts for the theory to b e dev elop ed in the next sections. Giv en 0 < q < 1 and a contin uous p oten tial A : { 1 , 2 , ..., d } → R , the q -pressure of β A , deﬁned in ( 1.6 ) and inv olving a non-extensive en tropy and an extensiv e integral, is deﬁned as P q ( β A ) = sup p n H q ( p ) + β d X j =1 p j a j o = sup p n 1 1 − q ( d X i =1 p q i − 1) + β d X j =1 a j p j o (4.1) where the supremum is taken ov er all probability v ectors p on { 1 , 2 , . . . , d } , and a prob- abilit y vector p is a q -e quilibrium for β A if p attains the supremum ab ov e. The classical pressure of β A is similar to the previous expression with H q ( p ) replaced by P d j =1 − p i log p j . It is easy to c heck (using ( 12.4 )) that P ( A ) ⩾ P q ( A ) for ev ery for 0 < q < 1 and that P ( A ) ⩽ P q ( A ) for ev ery q > 1. Lemma 4.1. Given q > 0 , β ∈ R and a p otential A = ( a 1 , a 2 , ..., a d ) , the q -e quilibrium state p = ( p 1 , p 2 , ..., p d ) for β A is unique and given by p j = e β a j 2 − q P d i =1 e β a i 2 − q , for every 1 ⩽ j ⩽ d . (4.2) Pr o of. In order to determine the probabilit y that attains the maximal v alue of ( 4.1 ) sub ject to the constraint P d j =1 p j = 1 we use Lagrange multipliers for the function (( p 1 , p 2 , . . . , p d ) , λ ) 7→ L (( p 1 , p 2 , . . . , p d ) , λ ) := β d X j =1 a j p j + 1 1 − q ( d X j =1 p q j − 1) − λ h d X j =1 p j − 1 i where λ is a constant. Given 1 ⩽ j ⩽ d the condition dL dp j ( p, λ ) = 0 can b e written as β a j + q 1 − q p q − 1 j − λ = 0 (4.3) A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 19 or, equiv alently , p j =  1 − q q  1 q − 1  λ − β a j  1 q − 1 =  1 − q q  1 q − 1 λ 1 q − 1  1 + β a j λ (1 − q ) ( q − 1)  1 q − 1 . sub ject to the constraint that P d j =1 p j = 1. T aking λ = 1 1 − q ab o v e we obtain ˜ p j =  1 q  1 q − 1  1 + β a j ( q − 1)  1 q − 1 =  1 q  1 q − 1 e β a j 2 − q . T aking the normalization we conclude that p j = ˜ p j P d i =1 ˜ p i = e β a j 2 − q P d i =1 e β a i 2 − q for eac h 1 ⩽ j ⩽ d . This prov es the lemma. □ R emark 4.2 . Given 0 < q < 1 and A = ( a 1 , a 2 ) , B = ( b 1 , b 2 ), after a tedious computation one can sho w that d dβ P q ( A + β B ) | β =0  = Z B dp A = b 1 e a 1 2 − q P 2 r =1 e a r 2 − q + b 2 e a 2 2 − q P 2 r =1 e a r 2 − q , (4.4) where p A is the q -equilibrium state for A = ( a 1 , a 2 ) . In this w ay , in the non-extensiv e case, the deriv ativ e of the pressure do es not b eha ve exactly like in the classical extensiv e case. In the sp ecial case that n = 2, q = 1 / 2 and A = ( a 1 , a 2 ) = (3 , 7), the graph of the function β → P q ( β A ) (see Figure 4.2 ) indicates that this function is neither concav e, nor con v ex, nor monotonous in the interv al ( − 0 . 5 , 1 . 5). This is due to the fact that the entrop y p → H q ( p ) may b e con v ex or conca ve dep ending on q (see Figure 1 ) and this somehow inﬂuence what is observ ed in the pressure when the v alue of β c hanges. Similarly , taking β = 1 and ﬁxed p oten tial A = ( a 1 , a 2 ) = (3 , 7) the graph obtained in Mathematica for q → P q ( A ) indicates that this function is neither concav e, nor conv ex, nor monotonous. Finally , in what follo ws we illustrate ho w Lemma 4.1 can b e used to compute the non-dynamical q -pressure function. Example 4.3 . If q = 1 / 3, n = 2, β = 1 . 2 , a 1 = 0 . 5 , a 2 = 0 . 8, q -pressure for β A is equal to 1 . 6895 .. and the q -equilibrium state p is p = ( p 1 , p 2 ) giv en by p 1 = 0 . 3172 ... = e β a 1 2 − q e β a 1 2 − q + e β a 2 2 − q and p 2 = 0 . 6828 ... = e β a 2 2 − q e β a 1 2 − q + e β a 2 2 − q . (4.5) A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 20 - 0.5 0.5 1.0 1.5 - 1 1 2 3 4 5 Figure 2. The graph of the function β → P q ( β A ), when β ∈ ( − 0 . 5 , 1 . 5), for q = 1 / 2, and A = ( a 1 , a 2 ) = (3 , 7), obtained in Mathematica. Example 4.4 . In the case q = 1 / 2 we get that the probabilit y vector ( p 1 , p 2 ) attaining the maxim um in Lemma 4.1 is given by p 1 = ( a 2 b − b ) 2 8 − 4 a 1 b − 4 a 2 b + a 2 1 b 2 + a 2 2 b 2 and p 2 = ( a 1 b − b ) 2 8 − 4 a 1 b − 4 a 2 b + a 2 1 b 2 + a 2 2 b 2 . 4.2. The dynamical q -pressure. Giv en 0 < q < 1, and a contin uous p otential A : Ω → R , the dynamic al q -pr essur e of A on G (or q -pr essur e for short, when no confusion is p ossible) is deﬁned as P G q ( A ) = sup µ ∈G n H q ( µ ) + Z A dµ o = sup µ ∈G n Z log q  1 J  dµ + Z A dµ o , (4.6) An y probabilit y measure µ ∈ G attaining the suprem um ( 4.6 ) is called a q -e quilibrium state asso ciated to A , for the q -pressure function in G . A diﬃcult y that arises in this context is that the set G is non-compact. W e deﬁne, analogously , for a contin uous potential A : Ω → R and 0 < q < 1, the dynamic al q -pr essur e of A on M inv ( σ ) is deﬁned by P M inv ( σ ) q ( A ) = sup µ ∈M inv ( σ ) n H q ( µ ) + Z A dµ o . (4.7) A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 21 An y p robability measure µ attaining the suprem um in ( 4.7 ) is referred to as a q -e quilibrium state asso ciated to A , for the q -pressure on the compact set M inv ( σ ). R emark 4.5 . Given 0 < q ⩽ 1, b y b oundedness and upp er semi-con tin uit y of the q -en trop y function (recall ( 3.1 ) and Lemmas 3.4 and 3.6 ) and compactness of M inv ( σ ), there alwa ys exist q -equilibrium states associated to the con tin uous potential A on M inv ( σ ). Moreo v er, ev en though G is a non-compact analytic manifold, the existence of a q -equilibrium state for A on G that attains the supremum in ( 4.6 ) follo ws by b oundedness, upp er semi- con tin uity and concavit y of the q -en trop y function. R emark 4.6 . The space G coincides with the space of extensiv e equilibrium states for Lipsc hitz contin uous p otentials. F or that reason, using Rohklin form ula ( 1.7 ), the classical pressure P ( A ) of a Lipschitz contin uous p otential A satisﬁes P ( A ) = sup µ ∈G n Z log J dµ + Z A dµ o , where J denotes the Jacobian asso ciated to µ . Moreo v er, from ( 3.1 ) w e get that P ( A ) ⩾ P q ( A ) for ev ery 0 < q < 1 and A ∈ C (Ω , R ) W e no w pro ceed to study the diﬀeren tiabilit y of the pressure function. Let us ﬁrst recall some necessary notions. W e sa y that B : Ω → R is c ohomolo gous to A : Ω → R if there exists a contin uous function f : Ω → R , and c ∈ R such that A = B + f − ( f ◦ σ ) − c (4.8) As G ⊂ M inv ( σ ), if ( 4.8 ) holds then R A dµ = R B dµ − c for every µ ∈ G and, consequently , the q -equilibrium states asso ciated with an y tw o cohomologous p otentials A and B are the same. In [ 15 ], Bi ´ s et al in tro duce an axiomatic deﬁnition of pressure function as any map Γ : B → R on a Banach space B ⊂ L ∞ (Ω) that satisﬁes the follo wing prop erties: for any A, B ∈ B and c ∈ R , (H1) (monotonicity) if A ⩽ B then Γ( A ) ⩽ Γ( B ); (H2) (translation in v ariance) Γ( A + c ) = Γ( A ) + c ; (H3) (conv exit y) Γ( αA + (1 − α ) B ) ⩽ α Γ( A ) + (1 − α )Γ( B ) for every 0 ⩽ α ⩽ 1. It is easy to chec k from its deﬁnition in ( 4.6 ) that P q : C 0 (Ω , R ) → R is a pressure function. Additionally , as the supremum in ( 4.6 ) is taken ov er probability measures that are σ -inv arian t then the pressure function P q also satisﬁes: (H4) (cob oundary inv ariance) P q ( A + f ◦ σ − f ) = P q ( A ) for ev ery A, f ∈ C 0 (Ω , R ). In view of [ 15 ] we obtain the following immediate consequence: Corollary 4.7. In the lo cus of c onvexity, the q -pr essur e function P q : Lip (Ω) → R is Gate aux diﬀer entiable if and only if it has a unique q -e quilibrium state on M inv ( σ ) for every Lipschitz c ontinuous p otential. A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 22 The next prop osition allows to obtain non-extensiv e equilibrium states from the classical extensiv e framework. Prop osition 4.8. L et µ ∈ G b e the extensive e quilibrium state for the p otential log J . Then, the fol lowing holds: (1) µ is a q -e quilibrium state for the p otential − log q ( 1 J ) ; (2) µ is the unique q -e quilibrium state for the p otential − log q ( 1 J ) which b elongs to G ; (3) P q ( − log q 1 J ) = 0 . . Pr o of. Fix µ ∈ G as the extensiv e equilibrium state for the p otential log J . Hence 0 = P (log J ) = H q ( µ ) + Z log J dµ = Z log q  1 J  dµ − Z log  1 J  dµ. No w, for each ˜ µ ∈ G , if one denotes its Jacobian by ˜ J , it follows from ( 3.8 ) that Z log q  1 J ( x )  d ˜ µ ( x ) − Z log q  1 ˜ J ( x )  d ˜ µ ( x ) ⩾ 0 (4.9) and, in particular, recalling ( 1.10 ), P q  − log q  1 J ( x )  = sup ˜ µ ∈G n H q ( ˜ µ ) + Z − log q  1 J ( x )  d ˜ µ ( x ) o = sup ˜ µ ∈G n Z log q  1 ˜ J ( x )  d ˜ µ ( x ) − Z log q  1 J ( x )  d ˜ µ ( x ) o ⩾ 0 . Moreo v er, the probabilit y measure ˜ µ = µ is the only measure in G for which the equality in ( 4.9 ) is attained. This implies simultaneously that P q  − log q  1 J ( x )  = 0 and that µ is the unique q -equilibrium state in G for the potential − log q  1 J ( x )  . This pro v es the prop osition. □ R emark 4.9 . W e observ e that while − log J = log  1 J  , the log q terms app earing in the non-extensiv e thermo dynamic formalism ob ey to a symmetry in parameters. More pre- cisely , if A = − log q ( 1 J ) then (observing the relation b etw een log q and e q in ( 12.7 )) log J = − log( e − A q ) = log ( e A 2 − q ) . (4.10) 5. Normalized potentials and eigenfunctions of transfer opera tors Giv en the potential A , we would like to determine the v alue of the q -dynamical pressure P q ( A ) in a pro cedure similar to that obtained when ﬁnding the eigenfunction of the Ruelle op erator (as in [ 62 ]). There are some tec hnical and conceptual diﬃculties in trying to implemen t an alik e strategy , due to the fact that e x + y q  = e x q e y q . F or that reason we need to in tro duce diﬀeren t and alternative concepts which are also natural to study . A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 23 5.1. Normalizable p oten tials. Deﬁnition 5.1 . Giv en q > 0 and a Lipschitz contin uous p oten tial A , we say that A is q -normalize d if d X a =1 e A ( ax ) q = 1 , for every x ∈ Ω . In case there exists a Lipschitz contin uous function φ A and a constan t c A suc h that d X a =1 e A ( ax )+ φ A ( ax ) − φ A ( x ) − c A q = 1 for every x ∈ Ω (5.1) then w e say that A is q -normalizable . It is clear from the deﬁnition that the p oten tial A = 0 is normalizable (there exists the trivial solution φ = 0 and c = log q ( d )). Moreov er, in the extensive con text, corresp onding to q = 1, equation ( 5.1 ) can b e written as L A ( e φ A )( x ) = d X a =1 e A ( ax )+ φ A ( ax ) = e − c A e φ A ( x ) for ev ery x ∈ Ω , (5.2) meaning that e φ A is an eigenfunction for the transfer op erator L A asso ciated to the eigen- v alue e − c A . In particular, in the extensive framework for every Lipsc hitz con tin uous p o- ten tial A the transfer op erator L A has a sp ectral gap on the space of Lipsc hitz contin uous observ ables, hence every Lipschitz contin uous p otential is normalizable. In the non-extensiv e the situation changes drastically . More precisely , if q  = 1 then e x + y +(1 − q ) x y q = e x q e y q for ev ery x, y in the domain of the q -exp onen tial (cf. ( 12.8 )) and consequen tly the solutions of ( 5.1 ) b ecome unrelated to eigen v alues of an y ˜ q -transfer operator. This is a ma jor obstruction to obtain to the construction of the non-extensiv e thermo dynamic formalism using the to ols dev elop ed in the extensive framework. R emark 5.2 . In Examples 8.5 and 8.4 we illustrate the fact that the existence of a pair ( φ A , c A ) satisfying ( 5.1 ) in the non-extensiv e setting can b e muc h more subtle than its extensiv e counterpart. Solutions to the normalization problem will b e pro duced via the implicit function theorem, and this will give also a new metho d for the solution of eigen- functions for transfer op erators in the classical extensive framew ork (w e refer the reader to Sections 7.1 and 7 for more details). This metho d has the adv an tage of showing that for eac h 0 < q < 1, the eigenfunction φ A and the constan t c A v ary diﬀerentially with the p oten tial A . The next lemma ensures that the solutions of ( 5.1 ) are related to solutions of cob ound- ary equations required for the analysis of the q -pressure problem, namely that under A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 24 some normalization condition, ev ery Lipschitz con tinuous p oten tial is cohomologous to a p oten tial of the form − log q ( 1 J ). Lemma 5.3. T ake 0 < q < 1 and let A : Ω → R b e a Lipschitz c ontinuous p otential for which ther e exists a Lipschitz c ontinuous function φ A and a c onstant c A such that d X a =1 e A ( ax )+ φ A ( ax ) − φ A ( x ) − c A 2 − q = 1 for every x ∈ Ω , and that al l summands ab ove ar e strictly p ositive. Then, ther e exist a Lipschitz c ontinuous p ositive Jac obian J : Ω → R , a Lipschitz c ontinuous function φ : Ω → R and c ∈ R so that − log q  1 J  = A ( x ) + φ ( x ) − φ ( σ ( x )) − c. (5.3) Pr o of. By assumption, the function J ( x ) = e A ( x )+ φ ( x ) − φ ( σ ( x )) − c 2 − q is a Lipschitz conti nuous Jacobian. Now, using that e x q = ( e − x 2 − q ) − 1 , the latter is equiv alen t to 1 J ( x ) = ( e A ( x )+ φ ( x ) − φ ( σ ( x )) − c 2 − q ) − 1 = e − ( A ( x )+ φ ( x ) − φ ( σ ( x )) − c ) q and so ( 5.3 ) holds. □ R emark 5.4 . The previous lemma ensures that if A is (2 − q )-normalizable with the pair ( φ A , c A ) then J ( x ) = e A ( x )+ φ A ( x ) − φ A ( σ ( x )) − c A 2 − q is a Jacobian. This, com bined with ( 12.25 ), ensures that e − log q ( 1 J ) 2 − q = J = e A + φ A − ( φ A ◦ σ ) − c A 2 − q and, consequently , − log q ( 1 J ) = A + φ A − ( φ A ◦ σ ) − c A . This prov es that − log q ( 1 J ) is cohomologous to A − c A and, b y prop erties (H2) and (H4) on the pressure function, it follo ws that: (i) P q ( − log q ( 1 J )) = P q ( A ) − c A , and (ii) both p oten tials ha v e the same q -equilibrium states. 5.2. Pro of of Theorem A . Giv en a Lipsc hitz con tin uous p otential A : Ω → R , de- note by φ and c the solutions of equation ( 2.5 ) and write J ( x ) = e A ( x )+ φ ( x ) − φ ( σ ( x )) − c 2 − q for the corresponding Jacobian. By Lemma 5.3 and Remark 5.4 there exists a Lipsc hitz con tin uous function φ and c ∈ R so that − log q  1 J  = A ( x ) + φ ( x ) − φ ( σ ( x )) − c and P  − log q  1 J  = P ( A ) − c Moreo v er, from Prop osition 4.8 , the classical equilibrium state for the p otential log( J ) is a q -equilibrium state for − log q  1 J  (hence for A as well). The reasoning ab o v e shows that c is unique. In order to complete the pro of of the theorem it remains to pro v e that P q ( A ) = c . As b efore, w e parameterize an arbitrary ˜ µ ∈ G b y its Jacobian ˜ J . By deﬁnition of q -pressure A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 25 and in v ariance of probability measures in G , P q ( A ) = sup ˜ µ ∈G h H q ( ˜ µ ) + Z A d ˜ µ ( x ) i = sup ˜ µ ∈G h Z log q ( 1 ˜ J ( x ) ) d ˜ µ ( x ) + Z A ( x ) d ˜ µ ( x ) i = sup ˜ µ ∈G h Z − ( A ( x ) + φ ( x ) − φ ( σ ( x )) − c ) d ˜ µ ( x ) + Z A d ˜ µ ( x ) i = c. This ﬁnishes the pro of of the theorem. □ The follo wing is a direct consequence of the pro of of Theorem A and the classical thermo dynamic formalism. Corollary 5.5. Given 0 < q < 1 and a (2 − q ) -normalizable Lipschitz c ontinuous p otential A , ther e exists a unique q -e quilibrium state µ q ,A for A in G . Mor e over, µ q ,A is exact, has exp onential de c ay of c orr elations for Lipschitz observables and it varies diﬀer entiably with r esp e ct to A . 5.3. The (2 − q ) -Ruelle operator equation. In this subsection we will relate solutions for the q -Ruelle op erator with solutions for the classical Ruelle op erator. Giv en 0 < q < 1 and a Lip chitz con tinuous p otential A : Ω → R , if for some φ A and c A w e get for any x ∈ Ω d X a =1 e A ( ax )+ φ A ( ax ) − φ A ( x ) − c A 2 − q = 1 , (5.4) then from Theorem A the v alue c A satisﬁes P q ( A ) = c A , R emark 5.6 . In case there exists a con tinuous function φ A and c A ∈ R satisfying the previous relation we say that q -Ruelle theorem equation can b e solv ed. If this is the case, from ( 2.7 ) a q -equilibrium state can b e given in terms of the solutions for the Ruelle equation whic h are known in the classical Thermo dynamic F ormalism. W e denote by L B the classical Ruelle op erator for the p oten tial B : Ω → R . Given an α -H¨ older contin uous function F : Ω → R , 0 < α ⩽ 1, we denote by | F | α = sup x,y ∈ Ω , x  = y | F ( x ) − F ( y ) | d ( x, y ) α its α -H¨ older constan t. It is not hard to c hec k that if f is Lipsc hitz con tin uous then log q ( f ) and e f q are Lipsc hitz contin uous and that | log q ( f ) | 1 ⩽ | f | 1 sup( | f | − q ) A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 26 and | e f q | 1 ⩽ 1 1 − q | f | 1 sup( | 1 + (1 − q ) f | q 1 − q ) . W e will need the next instrumen tal result. W e consider the following space of p otentials. Given q > 0 ( q  = 1), let us denote by H q the space of Lipschitz con tinuous functions A : Ω → R suc h that A ( x ) > 1 q − 1 , for all x ∈ Ω. (5.5) By compactness of Ω, the latter condition ( 5.5 ) ensures that (1 − q ) A is b ounded aw a y from − 1, hence it makes sense to deﬁne the following. Deﬁnition 5.7 . Given q > 0 with q  = 1 and a Lipschitz con tin uous p oten tial A ∈ H q consider the p oten tial A q : Ω → R given by A q = 1 1 − q log(1 + (1 − q ) A ) = log ( e A q ) . (5.6) Note that the p oten tial A q dep ends in a diﬀeren tiable fashion from A . Giv en a p oten tial A ∈ H q , the q -Ruel le op er ator L A,q : C 0 (Ω , R ) → C 0 (Ω , R ) is deﬁned b y f 7→ L A,q ( f ), where L A,q ( f )( x ) = X a e A ( ax ) q f ( ax ) = X a (1 + (1 − q ) A ( ax )) 1 1 − q f ( ax ) . (5.7) Lemma 5.8. Fix 0 < q < 1 and a p otential A ∈ H q . The fol lowing pr op erties hold: (1) the op er ators L A q and L A,q c oincide; (2) ther e exists a le ading eigenvalue λ A,q > 0 and a unique normalize d p ositive eigen- function φ A,q , in the sense that L A,q ( φ A,q ) = λ A,q φ A,q ; (5.8) (3) P ( A q ) = log λ A,q is the classic al pr essur e for the p otential A q , (4) P ( A ) ⩾ P ( A q ) . Pr o of. Note that, for any given contin uous function f : Ω → R and every x ∈ Ω, L A q ( f )( x ) = d X a =1 e A q ( ax ) f ( ax ) = d X a =1 e 1 1 − q log(1+(1 − q ) A ( a x )) f ( ax ) = d X a =1 (1 + (1 − q ) A ( ax )) 1 1 − q f ( ax ) = L A,q ( f )( x ) . A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 27 This prov es item(1). Items (2) and (3) are a direct consequence of (1) together with the classical Ruelle’s theorem for the transfer op erator L A q . Finally , denoting by µ A,q the classical equilibrium state for the p otential A q one kno ws that P ( A q ) = h ( µ A,q ) + Z A q dµ A,q . In particular, the classical v ariational principle together with the fact that 1 1 − q log(1 + (1 − q ) x ) < x for eac h 0 < q < 1 and x > 0, ensures that P ( A ) ⩾ h ( µ A,q ) + Z A dµ A,q ⩾ h ( µ A,q ) + Z 1 1 − q log(1 + (1 − q ) A ) dµ = P ( A q ) . This pro ves item (4) and completes the pro of of the lemma. □ Some comments are in order. First, note that equation ( 5.8 ) is quite diﬀeren t from ( 2.5 ), whic h inv olv es the (2 − q )-exp map. Note from Lemma 5.8 that giv en a general A , we deﬁne A q (via ( 5.6 )), for whic h there exists a con tinuous function φ A,q and c A ∈ R so that d X a =1 e A q ( ax )+log φ A,q ( ax ) − log φ A,q ( x ) − log λ A,q = 1 , for all x ∈ Ω . (5.9) Observing that ( 5.6 ) is equiv alen t to e A q ( x ) = e A ( x ) q one concludes that d X a =1 e A ( ax ) q e log φ A,q ( ax ) − log φ A,q ( x ) − c A,q = d X a =1 e A q ( ax ) e log φ A,q ( ax ) − log φ A,q ( x ) − c A,q = 1 . (5.10) Our strategy in the next subsections is to explore relations of the form e x 2 − q = e y q e z , for some x, y , z . 5.3.1. The c ase q = 1 / 2 . In the case q = 1 / 2 we are able to relate the extensiv e Ther- mo dynamic F ormalism to the non-extensive Thermo dynamic F ormalism. This in a sense that given a general p oten tial A (and the asso ciated A q as in ( 5.6 )) we will b e able to exhibit a p oten tial B such that one can express a solution φ B for the (2 − q ) equation d X a =1 e B ( ax )+ φ B ( ax ) − φ B ( x ) − c B 2 − q = 1 . (5.11) A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 28 This will b e achiev ed via ﬁnding the eigenfunction of the classical Ruelle op erator for the p oten tial A q (see ( 5.15 )). In this wa y one can pro duce examples getting a normalized equation for the non- extensiv e Ruelle op erator for the parameter 2 − q . In order to do that we claim that when q = 1 / 2, given r and a e r = e a q is equiv alent to a = − 2 ± 2 e r/ 2 . Belo w we will c ho ose the option corresp onding to a = − 2 + 2 e r/ 2 in our computations. W e will not present the details of all computations to deriv e the potential B ; just the ﬁnal expressions. Consider the function g given by g ( a 1 , a 2 , C , a ) = − 4 + (2 + a ) √ e a 1 − a 2 − C [2 − ( a 1 − a 2 − C )] (2 + a ) √ e a 1 − a 2 − C . (5.12) One can sho w that e g ( a 1 ,a 2 ,C,a )+( a 1 − a 2 − C ) 2 − q = 1 / 4(2 + a ) 2 e a 1 − a 2 − C = e a q e a 1 − a 2 − C . (5.13) F or the parameter (2 − q ), a p oten tial B and φ B as in ( 5.11 ) will b e derived from A (or, from the associated A q ): given the p otential A q expressed via ( 5.6 ), consider the asso ciated φ A,q and λ A,q = e c A,q obtained from Lemma 5.8 . Finally , tak e B = − 4 + (2 + A ) q φ A,q c A,q ( φ A,q ◦ σ ) [2 − log ( φ A,q ) + log ( φ A,q ◦ σ ) + c A,q )] (2 + A ) q φ A,q c A,q ( φ A,q ◦ σ ) , (5.14) φ B = log φ A,q and c B = c A,q . Then, giv en x we get from ( 5.13 ), ( 5.9 ) and ( 5.10 ) d X a =1 e A ( ax )+( φ A ( ax ) − φ A ( x ) − c A ) 2 − q = d X a =1 e A ( ax ) q e log φ A,q ( ax ) − log φ A,q ( x ) − c A,q = d X a =1 e A q ( ax ) e log φ A,q ( ax ) − log φ A,q ( x ) − c A,q = 1 . (5.15) In this w a y we sho w the existence of solutions for the (2 − q )-Ruelle theorem equation ( 5.11 ) from classical results for the p oten tial A q . Note that we can also obtain A from A q via A = e (1 − q ) A q − 1 1 − q . In this wa y we can b egin our reasoning taking some A q of our c hoice. Explicit extensive solutions of eigenfunction and eigenv alue for a family of p oten tials that dep ends on inﬁnite coordinates on Ω are presented in [ 19 ]. Therefore, taking as A q a A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 29 p oten tial described in [ 19 ] one gets solutions for the q -Ruelle op erator equation for a family of non trivial p otentials A on the non-extensive setting. Note also that from ( 5.14 ) w e get that the p oten tial A dep ends on a diﬀerentiable wa y from the p otential A . Remember that the p otential A q dep ends in a diﬀerentiable fashion A and vice versa. 5.3.2. The gener al c ase 0 < q < 1 . The pro cedure is similar to the previous one but the solution is not so explicit and simple. Deﬁne g by g ( a 1 , a 2 , C , a ) = e − a 1 ( e a 2+ c (1 + a − aq ) 1 q − 1 ( e a 1 − a 2 − c (1 + a − aq ) 1 q − 1 ) q q − 1 − e a 1 (1 + a 2 + c + a 1( q − 1) − a 2 q − cq q − 1 . Expression ( 12.24 ) is very helpful on this section. One can sho w that e g ( a 1 ,a 2 ,C,a )+( a 1 − a 2 − C ) 2 − q = e a q e a 1 − a 2 − C . (5.16) Giv en A , consider the p otential A q giv en by ( 5.6 ) and the asso ciated eigenfunction φ A,q and eigen v alue λ A,q = e c A,q obtained from Lemma 5.8 . Denote A = g (log φ A,q , log ( φ A,q ◦ σ ) , c A,q , A ) , and φ A = log φ A,q and c A = c A,q . Then, giv en x we get from ( 5.16 ) and ( 5.10 ), in the same w a y as in ( 5.15 ) d X a =1 e A ( ax )+ φ A ( ax ) − φ A ( x ) − c A 2 − q = 1 . (5.17) 6. Asymptoticall y sub-additive q -potentials and asymptotic pressure This section is devoted to the pro of of Theorem B . W e ﬁrst describ e the sequence of p oten tials app earing in the family of transfer op erators L n , for n ∈ N . 6.1. Sequences of sub-additive p oten tials and transfer op erators. Fix 0 < q < 1 and a Lip chitz contin uous p oten tial A : Ω → R . F or eac h n ∈ N and x ∈ Ω w e write S n A ( x ) = P n − 1 j =0 A ( σ j ( x )). F or each n ∈ N , the linear op erator L n deﬁned b y L n ( f )( x ) = X σ n ( y )= x e S n A ( y ) q f ( y ) = X σ n ( y )= x (1 + (1 − q ) S n A ( y )) 1 1 − q f ( y ) = X σ n ( y )= x e 1 1 − q log( 1+(1 − q ) S n A ( y ) ) f ( y ) . (6.1) A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 30 is p ositive and b ounded. F rom ( 6.1 ) it is natural to inv estigate the family of Lipschitz con tin uous functions φ n ( y ) = 1 1 − q log( 1 + (1 − q ) S n A ( y ) ) , n ∈ N , whic h we refer to as the family of q -p otentials asso ciate d to A (we omit the dep endence of the sequence on q and the potential A for notational simplicity). Lemma 6.1. If the p otential A is non-ne gative then the se quenc e ( φ n ) n ∈ N is sub-additive, i.e. φ m + n ⩽ φ m ◦ σ n + φ n for every m, n ∈ N . Pr o of. Given 0 < q < 1 and a, b > 0 one has that 1 1 − q log( 1 + (1 − q ) ( a + b ) ) ⩽ 1 1 − q log( 1 + (1 − q ) a ) + 1 1 − q log( 1 + (1 − q ) b ) . The previous expression can b e obtained by taking exp onential on b oth sides of the inequalit y and using that 1 + x + y ⩽ (1 + x )(1 + y ) for every x, y > 0. In consequence, ( 1 + (1 − q ) ( a + b ) ) 1 1 − q ( 1 + (1 − q ) a ) 1 1 − q ( 1 + (1 − q ) b ) 1 1 − q ⩽ 1 . (6.2) F rom ( 6.1 ) we get that φ n + m ⩽ φ n + ( φ m ◦ σ n ) for all m, n ∈ N , as desired. □ Lemma 6.2. Assume that A : Ω → R is a c ontinuous p otential. Then the se quenc e ( φ n ) n ∈ N is asymptotic al ly sub-additive, i.e. ther e exists a sub-additive se quenc e ( ψ n ) n ∈ N such that lim n →∞ 1 n ∥ φ n − ψ n ∥ ∞ = 0 . In p articular, given a σ -invariant and er go dic pr ob ability me asur e µ , lim n →∞ 1 n φ n ( x ) = inf n ⩾ 1 1 n Z φ n dµ for µ -a.e. x ∈ Ω . and the map M ( σ ) ∋ µ 7→ lim n →∞ 1 n R φ n dµ is upp er-semic ontinuous. Pr o of. In the case that inf x ∈ Ω A ( x ) ⩾ 0, Lemma 6.1 ensures that ( φ n ) n ∈ N is sub-additive and there is nothing to prov e. Assume now that inf x ∈ Ω A ( x ) < 0 and c ho ose the constan t c = − 2 inf x ∈ Ω A ( x ) > 0. Then one can write 1 1 − q log( 1 + (1 − q ) S n A )( y ) ) = 1 1 − q log( 1 + (1 − q ) S n ( A + c )( y ) ) (6.3) + 1 1 − q log h 1 + (1 − q ) S n A )( y ) 1 + (1 − q ) S n ( A + c )( y ) i . (6.4) A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 31 The sequence ψ n ( x ) = 1 1 − q log( 1 + (1 − q ) S n ( A + c )( y ) ) app earing in ( 6.3 ) is sub-additive, b y Lemma 6.1 . Moreov er, since 0 < q < 1 and log y ⩽ y for every y ⩾ 1 one can use 1 ⩽ 1 + (1 − q ) S n ( A + c )( y ) 1 + (1 − q ) S n A ( y ) ⩽ 1 + (1 − q ) cn 1 + (1 − q ) n inf x ∈ Ω | A ( x ) | ⩽ 4 for ev ery large n ⩾ 1, to b ound ( 6.4 ) in the following wa y 0 ⩽ 1 (1 − q ) log  1 + (1 − q ) S n ( A + c )( y ) 1 + (1 − q ) S n A )( y )  ⩽ 4 (1 − q ) . This implies that 1 n ∥ φ − ψ n ∥ ∞ ⩽ 4 (1 − q ) n tends to zero as n → ∞ , which prov es the ﬁrst assertion in the lemma. The latter, together with Kingman’s sub-additiv e ergo dic theorem (see [ 77 , Theorem 10.1]) implies that, for ev ery σ -in v arian t and ergodic probabilit y µ on Ω, lim n →∞ 1 n ψ n ( x ) = inf n ⩾ 1 1 n Z ψ n dµ, for µ -a.e. x ∈ Ω. (6.5) F urthermore, as the map M ( σ ) ∋ µ 7→ inf n ⩾ 1 1 n R ψ n dµ is the inﬁm um of contin uous maps, then it is upp er-semicon tinuous. Finally , the second and third claims in the lemma are direct consequences of the corresp onding statemen ts for the sub-additiv e sequence ( ψ n ) n ∈ N and the fact that lim n →∞ 1 n ∥ φ n − ψ n ∥ ∞ = 0 . □ 6.2. Pro of of Theorem B . Consider a Lipschitz con tin uous p otential A : Ω → R and 0 < q < 1. W e will ﬁrst show that for any given x 0 ∈ Ω, the limit lim n →∞ 1 n log L n (1)( x 0 ) exists and it is indep endent of x 0 . Lemma 6.3. F or e ach x 0 in Ω , the se quenc e ( 1 n log L n (1)( x 0 )) n ⩾ 1 is c onver gent. Pr o of. Fix x 0 in Ω and consider the sequence a n = log L n (1)( x 0 ) = log X σ n ( y )= x 0 (1 + (1 − q ) S n A ( y )) 1 1 − q . (6.6) Giv en x ∈ Ω and n ∈ N , w e denote by y x j,n , 1 ⩽ j ⩽ d n , the collection of p oints y ∈ Ω satisfying σ n ( y ) = x . It is not hard to c hec k that there exists a uniform constan t H > 0 so that L m (1)( y x 0 j,n ) L m (1)( x 0 ) ⩽ H (6.7) A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 32 for ev ery x 0 , m ⩾ 1 and 1 ⩽ j ⩽ d n . Using ( 6.2 ) and ( 6.7 ) w e deduce that L m + n (1)( x 0 ) = d m + n X j =1  1 + (1 − q )[ S m A ( y x 0 j,m + n ) + S n A ( σ m y x 0 j,m + n )]  1 1 − q = d m X k =1 d n X j =1  1 + (1 − q )[ S m A ( y y x 0 j,n k,m − n ) + S n A ( y x 0 j,n )]  1 1 − q ⩽ d m X k =1 d n X j =1 [(1 + (1 − q )( S m A ( y y x 0 j,n k,m − n ))] 1 1 − q [1 + (1 − q )( S n A ( y x 0 j,n ))] 1 1 − q = L m (1)( y x 0 j,n ) L n (1)( x 0 ) ⩽ H L m (1)( x 0 ) L n (1)( x 0 ) for every m, n ⩾ 1. Then, the sequence ( a n ) n ∈ N giv en by ( 6.6 ) satisﬁes the weakly sub- additiv e condition a m + n ⩽ a m + a n + log H for every m, n ⩾ 1 and, from [ 68 , Theorem 1.9.2 ], w e get that lim n →∞ a n n = inf n ⩾ 1 a n n . (6.8) This pro ves that the sequence ( 1 n log L n (1)( x 0 )) n ⩾ 1 is con vergen t, as desired. □ The next lemma ensures that the previous limit do es not depend on the initial point. Lemma 6.4. F or every x 0 , x 1 ∈ Ω the fol lowing holds: lim n →∞ 1 n log L n (1)( x 0 ) = lim n →∞ 1 n log L n (1)( x 1 ) Pr o of. It is w ell kno wn that the p oten tial A , b eing Lipsc hitz contin uous, satisﬁes the follo wing b ounded distortion prop ert y: there exists C > 0 such that | S n A ( y ) − S n A ( y ′ ) | ⩽ C dist( x, x ′ ) . (6.9) for an y n ⩾ 1, an y p oin ts x, x ′ ∈ Ω and an y paired pre-images y = y j,n ∈ σ − n ( x ) and y ′ = y ′ j,n ∈ σ − n ( x ′ ) (1 ⩽ j ⩽ d n ) in the same in v erse branch for σ n . Moreo v er, as Ω has ﬁnite diameter, from ( 6.9 ) there exists H > 0 such that (1 + (1 − q ) S n A ( y )) 1 1 − q (1 + (1 − q ) S n A ( y ′ )) 1 1 − q =  1 + (1 − q ) S n A ( y ) 1 + (1 − q ) S n A ( y ′ )  1 1 − q < H (6.10) for an y n ⩾ 1 and any paired pre-images y , y ′ . Using that | (1 + (1 − q ) r ) 1 1 − q − (1 + (1 − q ) s ) 1 1 − q | = (1 + (1 − q ) r ) 1 1 − q    1 −  1 + (1 − q ) s 1 + (1 − q ) r  1 1 − q    A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 33 for any r, s > 0 and denoting by y i , y ′ i the paired n th pre-images of x, x ′ , resp ectiv ely , one obtains that: given n ⩾ 1, x, x ′ ∈ Ω and the corresp onding paired pre-images y , y ′ , | L n (1)( x ) − L n (1)( x ′ ) | =    X σ n ( y )= x (1 + (1 − q ) S n A ( y )) 1 1 − q − X σ n ( y ′ )= x ′ (1 + (1 − q ) S n A ( y ′ )) 1 1 − q    ⩽    d n X i =1 (1 + (1 − q ) S n A ( y i )) 1 1 − q | 1 −  1 + (1 − q ) S n A ( y ′ i ) 1 + (1 − q ) S n A ( y i )  1 1 − q |    ⩽ (1 + H ) L n (1)( x ) . As x, x ′ are arbitrary w e conclude that 1 2 + H ⩽ L n (1)( x ′ ) L n (1)( x ) ⩽ 2 + H for ev ery n ⩾ 1 and every x, x ′ ∈ Ω. The conclusion of the lemma follows from the last inequalities and the conv ergence established in Lemma 6.3 . □ A t this p oint w e pro v ed that for an y 0 < q < 1, the q -asymptotic pressure of the Lipsc hitz contin uous p otential A : Ω → R can be computed b y the limit P q ( A ) = lim n →∞ 1 n log L n (1)( x 0 ) for an arbitrary p oin t x 0 ∈ Ω . In order to complete the pro of of Theorem B it remains to pro v e that the q -asymptotic pressure satisﬁes the following v ariational principle and that the suprem um can b e attained. Lemma 6.5. P q ( A ) = sup ν ∈M inv ( σ ) n h ( ν ) + lim n →∞ 1 n Z φ n dν o , (6.11) wher e h ( ν ) is the Kolmo gor ov-Shannon entr opy of ν . Pr o of. Fix x 0 ∈ Ω. F or eac h v alue n ∈ N consider the partition C n of Ω formed b y the collection of 2 n cylinders of size n in Ω. W e index the elemen ts of the partition C n b y corresp onding sets I n j , 1 ⩽ j ⩽ d n . F or each v alue n , and 1 ⩽ j ⩽ d n , w e get that σ n ( I n j ) = Ω and eac h I n j is an injectivit y domain for σ n and, for that reason, w e may denote b y y x j,n ∈ I n n the unique n th preimage of the p oin t x in I n j . No w, for each 1 ⩽ j ⩽ d n , pick z j,n ∈ I n j maximizing the function I n j ∋ y → φ n ( y ) = 1 1 − q log( 1 + (1 − q ) S n A ( y ) ). F rom ( 6.10 ) we get that ( 1 + (1 − q ) S n A ( z j,n ) ) 1 1 − q ( 1 + (1 − q ) S n A ( y x j,n ) ) 1 1 − q < H (6.12) A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 34 for any n , x ∈ Ω and 1 ⩽ j ⩽ d n . An argumen t similar to the one used in the pro of of Lemma 6.4 implies that 1 2 + H ⩽ L n (1)( x 0 ) P d n j =1 e sup y ∈ I n j φ n ( y ) ⩽ 2 + H for ev ery n ⩾ 1. By Lemma 6.2 there exists a sub-additive sequence ψ = ( ψ n ) n ⩾ 1 so that lim n →∞ 1 n ∥ φ n − ψ n ∥ ∞ = 0. Altogether we deduce that P q ( A ) = lim n →∞ 1 n log L n (1)( x 0 ) = lim n →∞ 1 n log d n X j =1 exp( sup y ∈ I n j ψ n ( y )) coincides with the top ological pressure of the sub-additiv e sequence of contin uous p oten- tials ( ψ n ) n ⩾ 1 (see e.g. (9) in [ 30 ]). Com bining the v ariational principle for sub-additive sequences of p oten tials (see e.g. [ 30 , equation (14)]) and the second assertion in Lemma 6.2 w e obtain ( 6.11 ), as desired. □ Finally , w e note that the fact that P q ( A ) = max ν ∈M inv ( σ ) n h ( ν ) + lim n →∞ 1 n Z φ n dν o , (6.13) (hence q -asymptotic equilibrium states alwa ys exist) is a direct consequence of the upp er- semicon tin uity of the Kolmogorov-Shannon entrop y map M inv ( σ ) ∋ ν 7→ h ( ν ) (see [ 77 ]) and the the upp er-semicon tinuit y of the map M inv ( σ ) ∋ ν 7→ lim n →∞ 1 n R φ n dν (recall Lemma 6.2 ). This ﬁnishes the pro of of Theorem B . □ 6.3. On the space of q -asymptotic equilibrium states. The family Φ = ( φ n ) n ∈ N of Lipschitz contin uous p oten tials is merely sub-additiv e in general. Nev ertheless, as it satisﬁes the v ariational principle ( 6.13 ) it mak es sense to ask whether the q -asymptotic equilibrium measures can b e derived from the classical extensive thermo dynamic formal- ism. W e start b y the follo wing: Lemma 6.6. The function C 0 (Ω) ∋ A 7→ P q ( A ) is a pr essur e function. Pr o of. The monotonicit y assumption (H1) is immediate from the deﬁnition of P q ( A ). The translation in v ariance and the conv exit y are immediate consequences of ( 6.13 ). □ In view of [ 15 , Lemma 8.3], we deﬁne the following measurable, upp er-semicon tin uous and b ounded p oten tial ψ A : Ω → R deﬁned by ψ A ( x ) = inf n ⩾ 1 1 n φ n ( x ) = inf n ⩾ 1 h 1 (1 − q ) n · log ( 1 + (1 − q ) S n A ( x ) ) i for every x ∈ Ω. W e pro v e the follo wing v ariational principle whic h inv olv es the usual a v erages of the p otential ψ A . A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 35 Prop osition 6.7. Ther e exists an upp er-semic ontinuous function h : M a ( σ ) → R such that P q ( A ) = sup ν ∈M a ( σ ) n h ( ν ) + Z ψ A dν o , wher e M a (Ω) stands for the sp ac e of σ -invariant ﬁnitely additive me asur es. In p articular, ther e exists a σ -invariant ﬁnitely additive e quilibrium state ν ∈ M a ( σ ) for σ with r esp e ct to the p otential ψ A . Pr o of. F or each 0 < q < 1, the sub-additive family of Lipsc hitz contin uous potentials Φ = ( φ n ) n ∈ N satisﬁes inf n ⩾ 1 1 n φ n ( x ) ⩾ 1 1 − q · [1 − 2(1 − q ) inf x ∈ Ω A ( x )] . Therefore, P q ( A ) > −∞ for ev ery b ounded p oten tial A . Then, it follows from the pro of of [ 15 , Theorem 8.4] that there exists an upp er-semicon tinuous en tropy function h : M a ( σ ) → R , deﬁned by h ( µ ) = inf ψ ∈ L ∞ (Ω)  P q ( A ) − R ψ A dµ  and satisfying P q ( A ) = sup ν ∈M a ( σ ) n h ( ν ) + Z ψ A dν o . Finally , the second claim follows as a direct consequence of the ﬁrst one together with the upp er semicontin uit y of the functions ν 7→ h ( ν ) and ν 7→ R ψ A dν . This completes the pro of of the prop osition. □ 7. Solution of Bowen-type equa tions for the non-extensive pressure and non-extensive transfer opera tors In this section we will study the space of normalizable potentials, which are related to the existence of eigenfunctions. 7.1. Existence of eigenfunctions for extensive transfer op erators. W e ﬁrst prov e the follo wing w arm-up theorem within the classical extensive framework (this corresp onds to the sp ecial case q = 1 in Theorem C ). Theorem 7.1. L et ˜ A : Ω → R b e a normalize d Lipschitz c ontinuous p otential and ν ˜ A b e such that L ∗ ˜ A ν ˜ A = ν ˜ A . Ther e exists an op en neighb orho o d U ⊂ Lip (Ω) of ˜ A and a diﬀer entiable map U ∋ A 7→ ( φ A , c A ) ∈ Lip (Ω) × R such that: (a) R φ A dν ˜ A = 0 , and (b) P d a =1 e A ( ax )+ φ A ( ax ) − φ A ( x ) − c A = 1 , for every x ∈ Ω . Mor e over, lim A → ˜ A ( φ A , c A ) = (0 , 0) ∈ Lip (Ω) × R . A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 36 7.1.1. Diﬀer entia bility of tr ansfer op er ators. Let ˜ A : Ω → R b e a ﬁxed normalized Lips- c hitz contin uous p otential and ν ˜ A b e a probabilit y measure such that L ∗ ˜ A ν ˜ A = ν ˜ A . Consider the analytic map F : Lip(Ω) × Lip(Ω) × R → Lip(Ω) × R ( A, φ, c ) 7→ ( P d a =1 e A ( a · )+ φ ( a · ) − φ ( · ) − c , R φ dν A ) . (7.1) and write F A ( · , · ) = F ( A, · , · ) for notational simplicity . Giv en A ∈ Lip(Ω), φ ∈ Lip(Ω), c ∈ R and ( H , h ) ∈ Lip(Ω) × R one can use the T a ylor expansion of the exp onen tial map to write F A ( φ + H , c + h ) − F A ( φ, c ) =  d X a =1 [ e A ( a · )+( φ + H )( a · ) − ( φ + H )( · ) − ( c + h ) − e A ( a · )+ φ ( a · ) − φ ( · ) − c ] , Z H dν A  =  d X a =1 e A ( a · )+ φ ( a · ) − φ ( · ) − c  e H ( a · ) − H ( · ) − h − 1  , Z H dν A  =  d X a =1 e A ( a · )+ φ ( a · ) − φ ( · ) − c  H ( a · ) − H ( · ) − h  , Z H dν A  + O ( ∥ H ∥ 2 ) + O ( h 2 ) , (as usual the terminology O ( y ) means that there exists B > 0 so that the expression is b ounded by B | y | ). The ﬁrst term ab o ve is linear in ( H , h ) while the second and third terms corresp ond to higher order terms. Hence, we conclude that the deriv ativ e D F A ( φ, c ) : Lip(Ω) × R → Lip(Ω) × R is given by D F A ( φ, c )( H , h ) =  d X a =1 e A ( a · )+ φ ( a · ) − φ ( · ) − c  H ( a · ) − H ( · ) − h  , Z H dν A  . (7.2) 7.1.2. Pr o of of The or em 7.1 . Let ˜ A : Ω → R be a ﬁxed normalized Lipschitz contin uous p oten tial and ν ˜ A b e a probability measure suc h that L ∗ ˜ A ν ˜ A = ν ˜ A . By the assumption, the equation d X a =1 e ˜ A ( ax )+ ˜ φ ( ax ) − ˜ φ ( x ) − ˜ c = 1 , (7.3) has a solution for the function ˜ φ ≡ 0 and the constant ˜ c = 0 (in this case ν ˜ A is σ -inv arian t). Equiv alently F ˜ A (0 , 0) = (1 , 0) . (7.4) Our purp ose is to use the formulation in ( 7.4 ) in order to use the implicit function theorem to show that there exists an op en neighborho o d of the p oten tial ˜ A formed by normalizable p oten tials. Prop osition 7.2. D F ˜ A (0 , 0) : Lip (Ω) × R → Lip (Ω) × R is an isomorphism. A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 37 Pr o of. Using that 1 is a simple leading eigenv alue and L ˜ A 1 = 1, one can write Ker( D F ˜ A (0 , 0)) = n ( H , h ) ∈ Lip(Ω) × R : d X a =1 e ˜ A ( ax ) ( H ( ax ) − H ( x )) = h, Z H dν ˜ A = 0 o . In particular, if ( H , h ) ∈ Ker( D F ˜ A (0 , 0)) then H = L ˜ A H − h and Z H dν ˜ A = Z L ˜ A H dν ˜ A − h. As L ∗ ˜ A ν ˜ A = ν ˜ A the second equation implies that h = 0. Consequently , all elements in the k ernel of D F ˜ A (0 , 0) are of the form ( H , 0) where L ˜ A H = H (hence H is constan t) and R H dν ˜ A = 0. This implies that DF ˜ A (0 , 0) is injective. W e pro ceed to sho w that D F ˜ A (0 , 0) is surjectiv e. Fix an arbitrary ( ψ , b ) ∈ Lip(Ω) × R . By deﬁnition, the equation D F ˜ A (0 , 0)( H , h ) = ( ψ , b ) admits a solution ( H , h ) if and only if ( L ˜ A ( H − H ◦ σ ) = ψ + h R H dν ˜ A = b (7.5) In tegrating with resp ect to ν ˜ A and noticing that we ha v e it is σ -inv arian t (b ecause ˜ A is normalized) the ﬁrst equation implies that h = − R ψ dν ˜ A . W e will now make use of the sp ectral theory for the transfer op erator L ˜ A . Denote by C = { ϕ − ϕ ◦ σ + z : ϕ ∈ Lip(Ω) , z ∈ R } (7.6) the subspace formed by cob oundaries and set C 0 = { ϕ − ϕ ◦ σ : ϕ ∈ Lip(Ω) } . It is kno wn that the linear map L ˜ A | C : C → Lip(Ω) is onto (cf. [ 41 , Prop osition 3.3 item 4]). W e claim that the linear operator V : C 0 → { ϕ ∈ Lip(Ω) : R ϕ dν ˜ A = 0 } given by V ( H ) = L ˜ A ( H − H ◦ σ ) is a bijection. As the injectivity follo ws as in the pro of of the injectivit y of D F ˜ A (0 , 0), it remains to prov e the surjectivity of V . In fact, b y surjectivit y of L ˜ A | C , for an y ϕ ∈ Lip(Ω) so that R ϕ dν ˜ A = 0, there exists H ∈ Lip(Ω) and z ∈ R so that ϕ = L ˜ A ( H − H ◦ σ + z ) = L ˜ A ( H − H ◦ σ ) + z . In tegrating with resp ect to the σ -inv ariant probabilit y measure ˜ ν A one obtains 0 = Z ϕ dν ˜ A = Z ( H − H ◦ σ ) d L ∗ ˜ A ˜ ν ˜ A + z , hence z = 0. This shows that V ( H − H ◦ σ ) = ϕ , and prov es the surjectivity of V . Therefore, the solution of ( 7.5 ) is obtained as the unique solution of the cohomological equation H − H ◦ σ = V − 1 ( ψ + h ) whic h satisﬁes R H dν ˜ A = b . This ﬁnishes the proof of the prop osition. □ A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 38 W e are now in a p osition to ﬁnish the pro of of Theorem 7.1 . In fact, in view of Prop osition 7.2 , the implicit function theorem (see e.g. [ 45 , page 9] or [ 5 , 64 ]) ensures that there exists an op en neighborho o d U ⊂ Lip(Ω) of ˜ A and a diﬀerentiable map U ∋ A 7→ ( φ A , c A ) ∈ Lip(Ω) × R suc h that d X a =1 e A ( ax )+ φ A ( ax ) − φ A ( x ) − c A = 1 for every x ∈ Ω and R φ A dν ˜ A = 0. This is to sa y that ˜ A admits an op en neighborho o d formed by nor- malizable p otentials. Finally , the limit lim A → 0 ( φ A , c A ) = (0 , 0) ∈ Lip(Ω) × R follows as an immediate consequence of the diﬀeren tiability result. This ﬁnishes the pro of of the theorem. □ R emark 7.3 . The previous arguments make use of some known results for the classical transfer operators. Nev ertheless, the argumen ts do not mak e use of iterates of the transfer op erators, which is one of the main obstructions to develop a non-extensiv e thermo dy- namic formalism. F or instance, in our pro of w e used the claim of Theorem 3.3 item (4) in [ 41 ], whic h w as obtained from prop erties of the op erator ( I − L A ) − 1 , where A is a H¨ older normalized p otential. In a general case, if this is true, the argument w orks. W e b eliev e that following this reasoning it is p ossible to derive pathological examples where one gets C 2 diﬀeren tiabilit y but not analyticity . Finally , one should notice that the argu- men ts concerning the pro of of the surjectivity resemble similar constructions app earing for p erturbation theory of leading simple eigenv alues (cf. [ 39 , 64 ]). 7.2. The space of extensiv e normalizable p oten tials. In this subsection we pro- ceed with the pro of of Theorem C in the con text of the non-extensive thermo dynamic formalism. T o each normalizable Lipsc hitz con tin uous p oten tial ˜ A one can asso ciate the normalized p oten tial B ( ˜ A ) = ˜ A + φ ˜ A − φ ˜ A ◦ σ − c ˜ A , (7.7) where P ( ˜ A ) = c ˜ A and e c ˜ A is the simple leading eigenv alue of the transfer op erator L ˜ A and h ˜ A = e φ ˜ A is a leading eigenfunction which, up to a multiplicativ e constan t, w e may assume to satisfy R h ˜ A dν ˜ A = 1. Moreov er, one has that b oth maps h : ˜ A → h ˜ A ∈ C 0 (Ω , R ) and c : ˜ A → c ˜ A ∈ R deﬁned for p oten tials in Lip(Ω) are C 1 -smo oth, and that, denoting b y E 0 the space of mean zero observ ables with resp ect to ν ˜ A , D h ( ˜ A ) H = h ˜ A · Z  ( I − L ˜ A | E 0 ) − 1 (1 − h ˜ A )  · H dν ˜ A , (7.8) and D c ( ˜ A ) H = c ˜ A · Z h ˜ A · H dν ˜ A = c ˜ A · Z H dµ ˜ A . (7.9) A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 39 for ev ery H ∈ Lip(Ω) (cf. [ 16 , Arxiv v ersion] or [ 55 ]). W e now deduce that the normalized p oten tial v aries smoothly with the original Lipschitz contin uous p otential. Corollary 7.4. The map B : Lip (Ω) → Lip (Ω) given by ( 7.7 ) is C 1 -smo oth and D B ( ˜ A ) H = H + Z  ( I − L ˜ A | E 0 ) − 1 (1 − h ˜ A )  · ( H − H ◦ σ ) dν ˜ A − c ˜ A · Z h ˜ A · H dν ˜ A for every H ∈ Lip (Ω) . Pr o of. It is immediate from ( 7.8 ) and the chain rule that, for each H ∈ Lip(Ω), D φ ( ˜ A ) H = D (log ◦ h )( ˜ A ) H = 1 h ( ˜ A ) D h ( ˜ A ) H = Z  ( I − L ˜ A | E 0 ) − 1 (1 − h ˜ A )  · H dν ˜ A (7.10) This, together with ( 7.9 ), implies that B is C 1 -smo oth and yields the formula for the deriv ative of B . □ The next theorem oﬀers a generalization of Theorem 7.1 , by establishing a solution for Bo w en’s equation for general normalizable potentials. Theorem 7.5. Fix 0 < q < 1 and assume that ˜ A is normalizable Ther e exists an op en neighb orho o d U ⊂ Lip (Ω) of ˜ A such that for every Lipschitz c ontinuous p otential A ∈ U ther e exists a solution ( φ A , c A ) such that d X a =1 e A ( ax )+ φ A ( ax ) − φ A ( x ) − c A = 1 , for al l x ∈ Ω (7.11) and R φ A dµ ˜ A = 0 . Mor e over, U ∋ A 7→ ( φ A , c A ) is C 1 -smo oth. Pr o of. If ˜ A is normalized the result follo ws from Theorem 7.1 and there is nothing to pro v e. Hence we assume that ˜ A is not normalized. Let φ ˜ A and c ˜ A b e such that the p oten tial B = ˜ A + φ ˜ A − φ ˜ A ◦ σ − c ˜ A is normalized, hence P d a =1 e B ( ax ) = 1 for all x ∈ Ω. Consider the analytic map F : Lip(Ω) × Lip(Ω) × R − → Lip(Ω) × R deﬁned by F ( A, φ, c )( x ) = d X a =1 e A ( ax )+[ φ ˜ A ( ax ) − φ ˜ A ( x ) − c ˜ A ]+ φ ( ax ) − φ ( x ) − c , Z φ dµ ˜ A ! and observ e that F ( ˜ A, 0 , 0) = (1 , 0) . In order to prov e Theorem 7.5 w e pro ceed to v erify the assumptions of the implicit function theorem. In order to do so, ﬁrst we compute the deriv ativ e of F with resp ect to A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 40 ( φ, c ) at ( ˜ A, 0 , 0). Note that d X a =1 e ˜ A ( ax )+[ φ ˜ A ( ax ) − φ ˜ A ( x ) − c ˜ A ]+ v ( ax ) − v ( x ) − ( c + α ) − d X a =1 e ˜ A ( ax )+[ φ ˜ A ( ax ) − φ ˜ A ( x ) − c ˜ A ] − c = d X a =1 e ( ˜ A ( ax )+[ φ ˜ A ( ax ) − φ ˜ A ( x ) − c ˜ A ] − c ) ( e v ( ax ) − v ( x ) − α − 1) = d X a =1 e B ( ax ) − c ( e v ( ax ) − v ( x ) − α − 1) for ev ery v ∈ Lip(Ω) and α ∈ R . In particular, taking the T a ylor series for the exp onen tial map one concludes that, for ( v , α ) ∈ Lip(Ω) × R , D ( φ,c ) F ( ˜ A, 0 , 0)( v , α )( x ) = d X a =1 e B ( ax )  v ( ax ) − v ( x ) − α  , Z v dµ ˜ A ! =  L B v ( x ) − v ( x ) − α , Z v dµ ˜ A  , where L B stands for the Ruelle op erator asso ciated with the normalized p otential B . W e claim that the latter is an isomorphism from Lip(Ω) × R onto Lip(Ω) × R . In fact, giv en ( f , β ) ∈ Lip(Ω) × R we aim to prov e that has a unique solution ( v , α ). L B v − v − α = f , Z v dµ ˜ A = β . (7.12) Using that L ∗ B µ ˜ A = µ ˜ A and integrating the ﬁrst equation ab o v e with resp ect to µ ˜ A one deduces that α = − R f dµ ˜ A . Th us, the ﬁrst equation in ( 7.12 ) b ecomes ( I − L B ) v = −  f − Z f dµ ˜ A  is a µ ˜ A -mean zero observ able. Since B is normalized, the op erator I − L B is in vertible on the subspace of Lipschitz functions with µ ˜ A -mean zero, and so there exists a unique solution u ∈ Lip(Ω) of the previous equation with R u dµ ˜ A = 0. Then, the pair ( v , α ) with v = u + β and α = − R f dµ ˜ A is the only solution of ( 7.12 ). This prov es that D ( φ,c ) F ( ˜ A, 0 , 0) is bijectiv e. Hence, by the implicit function theorem, there exist an op en neigh b orho o d U of ˜ A and a C 1 map A ∈ U 7→ ( φ ( A ) , c ( A )) such that F ( A, φ ( A ) , c ( A )) = (1 , 0). Finally , deﬁning φ A = φ ( A ) + φ ˜ A , and c A = c ( A ) + c ˜ A , w e obtain a mean zero solution for ( 7.11 ) as desired. This concludes the proof. □ Denote by N the set of normalizable Lipschitz contin uous p oten tials. Building ov er the previous results one can prov e the following: A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 41 Theorem 7.6. Every Lipschitz c ontinuous p otential A : Ω → R is normalizable. Pr o of. The space N is an op en subset of Lip(Ω) as a consequence of Theorem 7.5 . In order to pro ve the theorem it is enough to prov e that N is closed. Let ( A n ) n ⩾ 1 b e a sequence in N conv erging to A ∈ Lip(Ω) in the Lipschitz norm (recall ( 2.1 )). W e will show that A ∈ N , that is, that L A has a leading eigen v alue and eigenfunction. Denote b y W n : Ω × Ω → R , n ⩾ 1, an in v olution kernel for A n , with the normalization W n ( ... 1 , 1 , 1 | 1 , 1 , 1 ... ) = 1. (we refer the reader to [ 13 , 52 , 53 , 58 ] for the existence of inv olution kernels). In particular, the dual p otential A ∗ n : Ω → R , depending on negativ e co ordinates of the tw o-sided shift, is deﬁned by the equation A ∗ n ( y ) = A n ( x ) + W n ( ˆ σ ( y | x )) − W n ( y | x ) (7.13) where ( x | y ) ∈ Ω × Ω and ˆ σ stands for the tw o sided shift on Ω × Ω. F or eac h n ⩾ 1, denote by ν A ∗ n the leading eigenmeasure for the p oten tial A ∗ n . It is kno wn that the inv olution kernel W n is Lip c hitz in b oth v ariables [ 53 , Prop osition 7 in Section 5], that the dual p oten tial dep ends Lip chitz contin uously on the p oten tial [ 25 , Section 3] and that φ n ( x ) = Z e W n ( x,y ) dν A ∗ n ( y ) is an eigenfunction for the Ruelle op erator L A n . Up to extract some subsequence, we ma y assume without loss of generalit y that ( ν A ∗ n ) n ⩾ 1 is weak ∗ con v ergent to ν . Moreov er, let W denote the limit of the corresp onding inv olution kernels W n , guaranteed by the Arz ` ela-Ascoli theorem. Then, one concludes that φ = lim n →∞ φ n = lim n →∞ Z e W n ( · ,y ) dν n ( y ) = Z e W ( · ,y ) dν ( y ) is an eigenfunction for the Ruelle operator L A asso ciated to the leading eigen v alue 1. This pro v es that N is a closed set and completes the pro of of the theorem. □ 7.3. Solution of Bo w en-t yp e equations for non-extensiv e transfer op erators. Fix 0 < q < 1. Giv en a Lipsc hitz contin uous p oten tial A : Ω → R recall it is q - normalizable (resp. q -normalized) if there exist a Lipschitz contin uous function φ A and c A , suc h that, 2 X a =1 e A ( ax )+ φ A ( ax ) − φ A ( x ) − c A q = 1  resp. 2 X a =1 (1 + (1 − q ) A ( ax )) 1 / (1 − q ) = 2 X a =1 e A ( ax ) q = 1  (7.14) and all summands are strictly p ositiv e for all x ∈ Ω. The p oten tial A ≡ 0 is q -normalizable for all 0 < q < 1 as φ A = 0 and c A = log 2 − q (2) = − 1+2 − 1+ q q − 1 are solutions for the equation ( 7.14 ), and so A ≡ log 2 − q (2) is q -normalized. A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 42 Theorem 7.7. Given 0 < q < 1 and ˜ A ∈ N q ther e exists an op en neighb orho o d U of ˜ A in Lip (Ω) such that for every A ∈ U ther e exists a pr ob ability me asur e ν ˜ A , a Lipschitz c ontinuous function φ A and c A ∈ R such that d X a =1 e A ( ax )+ φ A ( ax ) − φ A ( x ) − c A q = 1 for al l x ∈ Ω , and R φ A dµ ˜ A = 0 . Mor e over, φ A , c A dep end in a diﬀer entiable way on A . Pr o of. Let us prov e the result in the case A ∈ N q is normalized (the pro of on the general case can b e adapted in a natural w a y , b y deﬁning a similar map as in ( 7.11 ) with the exp onen tial terms replaced b y q -exp onential terms). Consider the analytic map F : Lip(Ω) × Lip(Ω) × R → Lip(Ω) × R , given by F ( A, φ, c )( x ) =  d X a =1 e A ( ax )+ φ ( ax ) − φ ( x ) − c q , Z ( φ − φ ◦ σ − c ) dµ ˜ A  , (7.15) for a ﬁxed probability measure µ ˜ A to b e determined later. The assumptions ensure that F ( ˜ A, 0 , 0) = (1 , 0). W e will obtain functions A 7→ φ ( A ) , A 7→ c ( A ) suc h that F ( A, φ ( A ) , c ( A )) = (1 , 0) as a consequence of the implicit function theorem. Observ e that the second co ordinate in the righ t-hand side of ( 7.15 ) is linear in ( φ, c ). Moreo v er, using the deriv ativ e of q -exp maps in ( 12.13 ), for each ( H , h ) ∈ Lip(Ω) × R , d X a =1 e A ( ax )+ H ( ax ) − H ( x ) − h q − d X a =1 e A ( ax ) q = d X a =1  e ˜ A ( ax )+ H ( ax ) − H ( x ) − h q − e ˜ A ( ax ) q  = d X a =1  1 + (1 − q ) ˜ A ( ax )  q 1 − q ( H ( ax ) − H ( x ) − h ) + O ( ∥ H ∥ 2 + | h | 2 ) , where the ﬁrst term in the right-hand side ab ov e is linear in ( H , h ). This pro ves that D F A (0 , 0)( H , h ) =  d X a =1  1 + (1 − q ) ˜ A ( a · )  q 1 − q ( H ( a · ) − H ( · ) − h ) , Z ( H − H ◦ σ − h ) dµ ˜ A  . (7.16) In view of ( 7.16 ) it is natural to consider the extensiv e non normalized potential q ˜ A q giv en by x 7→ q ˜ A q ( x ) = log ( [1 + (1 − q ) ˜ A ( x )] q 1 − q ) , (recall ( 5.6 )) and the asso ciated extensive Ruelle op erator L q ˜ A q ( f )( x ) = d X a =1 [1 + (1 − q ) ˜ A ( a x )] q 1 − q f ( ax ) . A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 43 Denote by µ ˜ A the extensive equilibrium state for the p oten tial q ˜ A q . By σ -inv ariance we deduce from ( 7.16 ) that D F ˜ A (0 , 0)( H , h ) =  L q ˜ A q ( H − H ◦ σ − h ) , − h  . The same argumen t as in the pro of of Prop osition 7.2 ensures that the linear map V : C 0 → Lip(Ω) 0 giv en by V ( H ) = L q ˜ A q ( H − H ◦ σ ) is an isomorphism. It is simple to chec k that the latter implies that D F ˜ A (0 , 0) is an isomorphism as w ell. Hence, the conclusion of the theorem is a direct consequence of the implicit function theorem. □ 8. Examples 8.1. One-step p oten tials. W e will consider ﬁrst the case of lo cally constant p otentials A dep ending just on the ﬁrst co ordinate. Example 8.1 . Consider the p oten tial A : { 1 , 2 } N → R , suc h that, is constant equal to a 1 on the cylinder 1 and equal to a 2 on the cylinder 2 . Also consider another p oten tial B : { 1 , 2 } R → R , such that, is constan t equal to b 1 on the cylinder 1 and equal to b 2 on the cylinder 2 . F or q = 3 / 2, we wan t to consider the equation ( 2.5 ) for the family of p oten tials A + sB , where s ∈ R . More precisely , we will exhibit explicit solutions, which will b e denoted by c ( s ) and φ s : { 1 , 2 } R → N , with a dep endence on the parameter s , for the equation 2 X a =1 e ( A + sB )( ax )+ φ s ( ax ) − φ s ( x ) − c ( s ) 2 − 3 2 = 1 , for any x = ( x 1 , x 2 , ... ) ∈ Ω. (8.1) Giv en s ∈ R , consider c ( s ) = 1 / 2(4 + a 1 + a 2 + b 1 s + b 2 s ). F rom Theorem A w e get that c ( s ) = P 3 / 2 ( A + s B ) . Consider the function φ s : { 1 , 2 } N → R such that for each s , it is constan t equal to φ 2 ( s ) on the cylinder 1 and equal to φ 1 ( s ) = 0 on the cylinder 2 , where φ 2 ( s ) = 1 / 2( − a 1 + a 2 − b 1 s + b 2 s + q 16 − a 2 1 + 2 a 1 a 2 − a 2 2 − 2 a 1 b 1 s + 2 a 2 b 1 s + 2 a 1 b 2 s − 2 a 2 b 2 s − b 2 1 s 2 + 2 b 1 b 2 s 2 − b 2 2 s 2 ]) . W e leav e to the reader the task to v erify that the equation ( 8.1 ) holds. When a 1 = 2, a 2 = 5 . 5, b 1 = 0 = b 2 , w e get φ 2 (0) = 2 . 71825 and c (0) = 5 . 75. The explicit expressions w ere obtained using the softw are Mathematica. In this wa y we get an explicit solution for ( 8.1 ), when s = 0. Note also that c (0) = P 3 / 2 ( A ) = sup p n H 3 / 2 ( p ) + Z A dp o . A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 44 F rom Prop osition 4.8 , if the previous A satisﬁes A = − log q ( 1 J ), w e get that P q ( A ) = 0 and the q -equilibrium state for A is the classical one for log J . It is well known that this one is the indep enden t probability p on Ω with weigh ts p 1 , p 2 , where p k = e α k 2 − q P 2 r =1 e α r 2 − q , for k = 1 , 2. (8.2) In fact, giv en the real num b ers α 1 and α 2 , w e get that e α 1 2 − q P 2 r =1 e α r 2 − q + e α 2 2 − q P 2 r =1 e α r 2 − q = 1 , and the summands are diﬀerent if α 1  = α 2 . Then consider the Jacobian J : { 1 , 2 } R → R , suc h that, it is constant equal to e α 1 2 − q P 2 r =1 e α r 2 − q on the cylinder 1 and equal to e α 2 2 − q P 2 r =1 e α r 2 − q on the cylinder 2 , and tak e a 1 = − log q ( 1 e α 1 2 − q P 2 r =1 e α r 2 − q ) and a 2 = − log q ( 1 e α 1 2 − q P 2 r =1 e α r 2 − q ) . In case q = 3 / 2, one obtains p 1 = (2 + α 2 ) 2 8 + 4 α 1 + α 2 1 + 4 α 2 + α 2 2 and p 2 = (2 + α 1 ) 2 8 + 4 α 1 + α 2 1 + 4 α 2 + α 2 2 . T aking deriv ativ e at s = 0, w e get d ds P 3 / 2 ( A + s B ) | s =0 = 1 2 b 1 + 1 2 b 2  = b 1 p 1 + b 2 p 2 = Z B d p . (8.3) R emark 8.2 . Expression ( 8.3 ) sho ws that the deriv ative of the non-extensiv e pressure diﬀers from the one in the extensive setting (cf. Prop osition 4.10 in [ 62 ]). In the App endix Section 9 we consider for the case q = 1 / 2 the deriv ativ e of pressure for a more general class of p oten tials (see ( 9.2 )). 8.2. Lo cally constan t t w o-step p oten tials. Now, we will consider the next lev el of complexit y considering the Ruelle op erator equation for p otentials A : { 1 , 2 } N → R whic h dep end just on the ﬁrst tw o co ordinates, that is, A ( x 1 , x 2 , x 3 , .., x n , ... ) = A ( x 1 , x 2 ) := a x 1 x 2 (8.4) whic h is constan t and equal to a ij in eac h cylinder i j , i, j = 1 , 2. The information of the p oten tial A is describ ed by the matrix A =  a 1 1 a 1 2 a 2 1 a 2 2  . (8.5) Giv en 0 < q < 1 w e wan t to ﬁnd φ and c , whic h are solutions of 2 X a =1 e A ( ax )+ φ ( ax ) − φ ( x ) − c q = 1 , (8.6) A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 45 with φ ( x 0 ) = α , for ﬁxed x 0 and α (we will refer to this setting as the Marko v case). W e will assume, without loss of generalit y , that A (1 , 1) = a 11 is the minimum of the en tries of the matrix ( 8.5 ). As a p oten tial A admits φ and c as solutions of ( 8.6 ) is equiv alent to sa y that ˆ A = A − A (1 , 1) has φ and ˆ c = c + A (1 , 1) as solutions of the same equation, in order to solve ( 8.6 ) w e will assume throughout that A (1 , 1) = a 11 = 0 . In this case all other entries of the ab ov e matrix are nonegative. W e will sho w examples of p oten tials of the form ( 8.4 ) so that the solutions φ are constan t equal to φ 1 in the cylinder 1 and equal to φ 2 in the cylinder 2. In this case, it follo ws from Prop osition 4.8 that the corresponding non-extensive equilibrium probability measure will b e a (classical) stationary Marko v probability . Observe also that if φ is a solution of ( 8.6 ), then adding a constan t to φ we also get a solution of ( 8.6 ). Therefore, it is enough to search for a solution φ whic h is equal to φ 1 = 0 in the cylinder 1 and equal to φ 2 in the cylinder 2. In this w a y the function φ dep ends on the ﬁrst co ordinate in Ω and is determined by the v alue φ 2 (it is equal to zero in the cylinder 1). R emark 8.3 . The assumption that all summands in ( 8.6 ) are strictly p ositive is necessary for the connection with the existence of p ositive Jacobians whic h are of great imp ortance in classical Thermo dynamic F ormalism. This issue is clearly related to the claims presented in Lemma 5.3 which in some w a y connects the non-extensiv e setting with the extensive setting. In consonance with the previous discussion w e will assume that all en tries in the matrix  e a 11 − c q e a 12 − φ 2 − c q e a 21 + φ 2 − c q e a 22 − c q  (8.7) are p ositiv e n umbers and that a 11 = 0, and consider the functions f 1 ( A, c, φ 2 ) = e − c q + e a 21 + φ 2 − c q and f 2 ( A, c, φ 2 ) = e a 12 − φ 2 − c q + e a 22 − c q . (8.8) The next example shows that the solutions φ of ( 8.6 ) need not be unique. Example 8.4 ( Non-uniqueness of solutions ) . Set a 11 = 0 = a 22 , q = 1 / 2 in ( 8.8 ). Assume φ is equal to φ 1 = 0 in the cylinder 1 and equal to φ 2 in the cylinder 2. T ak e c = 1 / 2(4 + q 16 − a 2 12 + 2 a 12 a 21 − a 2 21 ) (8.9) φ 2 = − 2 − a 21 + c + √ 4 c − c 2 , (8.10) when 4 c − c 2 ⩾ 0 and 16 − a 2 12 + 2 a 12 a 21 − a 2 21 ⩾ 0. One can show that f 1 ( A, c, φ 2 ) = 1 and f 2 ( A, c, φ 2 ) = 1 . (8.11) In this case w e get φ and c which are explicit solutions for the ˜ q = 1 / 2 non-extensiv e Ruelle Theorem. A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 46 When a 12 = 2 and a 21 = 3 . 5 we get that φ 2 = − 0 . 89595 and c = 3 . 85405 solv es ( 8.11 ). In this w ay we get explicitly φ and c solving ( 8.6 ). Surprisingly , if w e take ab ov e φ 2 = − 2 − a 21 + c − √ 4 c − c 2 (8.12) instead of ( 8.10 ), we get φ 2 = − 2 . 39595 and c = 3 . 85405 solving ( 8.6 ), for a 12 = 2 and a 21 = 3 . 5. In b oth cases, all v alues e A ( ax )+ φ ( ax ) − φ ( x ) − c q in ( 8.6 ) are p ositiv e. Therefore, it is p ossible to get tw o diﬀeren t eigenfunctions asso ciated to tw o diﬀeren t eigen v alues whic h satisfy the prop erty that all summands in ( 8.6 ) are p ositive. This sho ws that the non-extensive analogous of the Ruelle theorem displays issues of a muc h more complex nature. The next example will show that there are examples of p oten tials A whic h admit solu- tions for ( 8.6 ) where one of the summands is equal to zero. This is a further evidence that results analogous to the classical Ruelle’s Theorem, ab out the existence of eigenfunctions and eigen v alues in the non-extensive setting b ecomes a muc h more complex matter. No w we presen t explicit solutions for the eigenfunction problem cov ering diﬀeren t p os- sibilities. Example 8.5 . T ake 0 < q < 1 and a 11 = 0 = a 21 . W e will exhibit explicit form ulas for solutions c and φ for the system ( 1 = e − c q + e φ 2 − c q 1 = e a 12 − φ 2 − c q + e a 22 − c q . where all summands are p ositiv e. This will pro vide solutions φ for ( 8.1 ). Denote q 1 = e − c q and q 2 = e a 22 − c q , where we assume that the parameters are such that q 1 , q 2 ∈ (0 , 1). In terms of the v alues q 1 , q 2 , w e get the solutions c = − log q ( q 1 ) , (8.13) φ 2 = log q (1 − q 1 ) + c, (8.14) a 12 = log q (1 − q 2 ) + φ 2 + c, (8.15) and a 22 = log q ( q 2 ) + c. (8.16) The eigenfunction φ is equal to zero in the cylinder 1 and is equal to φ 2 in the cylinder 2 . F or example, when q = 2 / 3, a 12 = 0 . 857533, a 22 = 0 . 52199, w e get the q -eigenv alue c = 0 . 991701 and the q -eigenfunction φ , is suc h that φ 2 = 0 . 655413. This case corresp onds to q 1 = 0 . 3 and q 2 = 0 . 6. When q = 4 / 5, taking q 1 = 0 . 2 and q 2 = 0 . 3, w e get a 12 = 2 . 18972, a 22 = 0 . 306117, φ 2 = 1 . 15786 and c = 1 . 3761. In this wa y w e get another example where we can ﬁnd an explicit solution φ and c . F urthermore, when q = 1 / 2, a 11 = 0 = a 21 , a 12 = − 5 . 67332, a 22 = − 3 . 09545, one has that c = 0, and φ 2 = − 2 provide solutions φ for ( 8.1 ), and one has that e φ − c 1 / 2 = 0 . A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 47 9. Appendix A: The deriv a tive of the d ynamical q -pressure Recall that for eac h 0 < q < 1 and every Lipschitz con tinuous p oten tial A : Ω → R , the q -pressure of A is deﬁned b y P q ( A ) = sup µ ∈G h H q ( µ ) + Z A ( x ) dµ ( x ) i , where H q ( µ ) is the q -entrop y of µ ∈ G deﬁned in ( 1.10 ). F rom Remark 4.5 , for ﬁxed 0 < q < 1, we kno w that P q ( A ) is diﬀeren tiable in A . W e consider in this app endix inﬁnitesimal v ariations of a normalized p otential A (for the diﬀeren tiabilit y of the pressure function on the extensive case see e.g. [ 16 , 44 , 59 ]). Giv en the Lipschitz con tin uous function v : Ω → R w e wan t to compute d ds P q ( A + sv ) | s =0 for some q  = 1. W e will prov e the following lemma. Lemma 9.1. Consider a Lipschitz c ontinuous Jac obian J and the asso ciate d e quilibrium state µ and let A b e the normalize d p otential A = − log q ( 1 J ) . Assume that v is a Lipschitz c ontinuous p otential and assume that ther e exist ( φ s , c s ) such that d X a =1 e ( A ( ax )+ sv ( ax ))+[ φ s ( ax ) − ( φ s ( x )] − c s 2 − q = 1 (9.1) for every x ∈ Ω and every smal l s . If q = 1 2 then d ds P q ( A + sv ) | s =0 = R J ( x ) 1 / 2 v ( x ) dµ + R J ( x ) 1 / 2 ( d ds φ s ( x ) − d ds φ s ( σ ( x ))) | s =0 dµ R J ( x ) 1 / 2 dµ ( x ) . (9.2) Pr o of. Fix q = 1 / 2 and let A, v : Ω → R b e Lipschitz contin uous as in the statement. As e A 2 − q = J (recall ( 12.25 )) then P d a =1 e A ( ax ) 2 − q = P d a =1 J ( ax ) = 1 . In consequence, Z d X a =1 J ( ax ) f ( ax ) dµ = Z d X a =1 e A ( ax ) 2 − q f ( ax ) dµ = Z f dµ (9.3) for ev ery con tin uous function f and the q -pressure of A is P q ( A ) = 0 (cf. Proposition 4.8 ). Moreo v er, it is not hard to chec k that e f ( s )+ g ( s ) − h ( s ) 2 − q = h 1 − 1 2 ( f ( s ) + g ( s ) − h ( s )) i − 2 (9.4) and, using the formula for the deriv ativ e of the (2 − q )-exp map in ( 12.26 ), d ds e f ( s )+ g ( s ) − h ( s ) 2 − q = h 1 − 1 2 ( f ( s ) + g ( s ) − h ( s )) i − 3 · h f ′ ( s ) + g ′ ( s ) − h ′ ( s ) i . (9.5) A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 48 No w w e note that if ( 9.1 ) holds then c s = P q ( A + sv ) . Therefore, diﬀerentiating b oth sides of ( 9.1 ) with resp ect to the parameter s and using ( 9.4 ) and ( 9.5 ) 0 = d ds d X a =1 e ( A ( ax )+ sv ( ax ))+[ φ s ( ax ) − φ s ( x )] − c s 2 − q | s =0 = d X a =1  1 + 1 2 log q 1 J ( ax )  − 3 h v ( ax ) + ( d ds φ s ( ax ) − d ds φ s ( x )) | s =0 − d ds c s | s =0 i As (1 + 1 2 log q (1 /y ) ) − 1 = y 1 / 2 for an y y > 0 one can write  1 + 1 2 log q 1 J ( ax )  − 3 = J ( ax ) 3 / 2 and, consequen tly , d X a =1 J ( ax ) h J ( ax ) 1 / 2 v ( ax ) + J ( ax ) 1 / 2 ( d ds φ s ( ax ) − d ds φ s ( x )) | s =0 − J ( ax ) 1 / 2 d ds c s | s =0 i = 0 for ev ery x ∈ Ω . F urthermore, using that e A 2 − q = J , d X a =1 e A ( ax ) 2 − q [ J ( ax ) 1 / 2 v ( ax ) + J ( ax ) 1 / 2 ( d ds φ s ( ax ) − d ds φ s ( x )) | s =0 − J ( ax ) 1 / 2 d ds c s | s =0 ] = 0 . (9.6) In tegrating with resp ect to µ together with ( 9.3 ) and ( 9.6 ) one deduces that 0 = Z d X a =1 e A ( ax ) 2 − q [ J ( ax ) 1 / 2 v ( ax ) + J ( ax ) 1 / 2 ( d ds φ s ( ax ) − d ds φ s ( x )) | s =0 − J ( ax ) 1 / 2 d ds c s | s =0 ] dµ ( x ) = Z J ( x ) 1 / 2 v ( x ) + J ( x ) 1 / 2 ( d ds φ s ( x ) − d ds φ s ( σ ( x ))) | s =0 − J ( x ) 1 / 2 d ds c s | s =0 dµ ( x ) Z J ( x ) 1 / 2 v ( x ) dµ + Z J ( x ) 1 / 2 ( d ds φ s ( x ) − d ds φ s ( σ ( x ))) | s =0 dµ − d ds c s | s =0 Z J ( x ) 1 / 2 dµ ( x ) . This ﬁnishes the pro of of the lemma. □ 10. Appendix B: Shannon appro a ch to the non-extensive measure theoretic entrop y In this subsection w e shall introduce an alternative notion of non-extensive entrop y for all probabilit y measures in M ( σ ), whic h is a natural extension of the concept introduced b y M´ eson and V ericat [ 60 ] for Bernoulli probability measures. Fix 0 < q < 1. Given a A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 49 ﬁnite partition P of a compact metric space X , deﬁne H q ( µ, P ) = 1 1 − q  X P ∈P µ ( P ) q − 1  = 1 1 − q  e log  P P ∈P µ ( P ) q  − 1  , (10.1) whic h, using ( 1.3 ) one can write alternatively H q ( µ, P ) = X P ∈P µ ( P ) log q  1 µ ( P )  Lemma 3.8 ensures that H q ( µ, P ) ⩽ log q ( d ) and that its maximal v alue is attained in the case of the equidistributed Bernoulli probabilit y measure. Moreo v er, if P ( n ) = W n − 1 j =0 σ − j ( P ) denotes the dynamically deﬁned partition, H q ( µ, P ( n ) ) need not b e sub- additiv e for q  = 1, due to the prop erties of log q ( · ). Using ( 10.1 ) we obtain log[ 1 + (1 − q ) H q ( µ, P ) ] = log  X P ∈P µ ( P ) q  . (10.2) Inspired b y ( 10.2 ), M´ eson and V ericat deﬁne the h q -entr opy of µ by h q ( µ ) = sup P h q ( µ, P ) where the suprem um is taken ov er all ﬁnite partitions of X and h q ( µ, P ) = lim sup n →∞ 1 n log[ 1 + (1 − q ) H q ( µ, P ( n ) ) ] . (10.3) W e will need some auxiliary results. Lemma 10.1. The function ( p 1 , p 2 , . . . , p n ) 7→ log  P d i =1 p q i  deﬁne d on the sp ac e of pr ob ability ve ctors p = ( p 1 , p 2 , . . . , p n ) , and attains the maximum value 1 1 − q log( n ) at the e quidistribute d pr ob ability ve ctor. Pr o of. In order to make use Lagrange m ultipliers consider the function p = ( p 1 , p 2 , . . . , p n ) → log  d X j =1 p q j  − α d X j =1 p j where α is a constant. T aking deriv ativ e in p i , i = 1 , 2 , ..., n , we lo ok for the condition q p q − 1 i P d j =1 p q j − α = 0 , (10.4) whic h implies that p i = h α 1 q d X j =1 p q j i 1 q − 1 , ∀ i = 1 , 2 , ..., n. A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 50 This, together with the assumption that P d i =1 p i = 1, this implies that p i = 1 n , that p is the equidistributed probability vector and that the maximum v alue is log  P d i =1 n − q  . This pro ves the lemma. □ One could ask whether the usual notion of top ological entrop y admits a natural coun- terpart in the non-extensive framew ork. Given a ﬁnite op en cov ering U of X w e write N q ( U ) = log q (# U ). A simple mo diﬁcation of the arguments in [ 77 , Section 7.1] and using prop ert y ( 12.2 ) of q -log functions w e conclude that N q ( U ∨ V ) = log q ( U ∨ V ) = N q ( U ) + N q ( V ) + (1 − q ) N q ( U ) N q ( V ) for every ﬁnite open co v erings U , V of X . In consequence, denoting b y U ( n ) = W n − 1 j =0 σ − j ( U ) the dynamically deﬁned op en cov ering of U , one concludes that N q ( U n + m ) = log q  n − 1 _ j =0 σ − j ( U ) ∨ m − 1 _ j =0 σ − j ( σ − n ( U ))  ⩽ N q ( U ( n ) ) + N q ( U ( m ) ) + (1 − q ) N q ( U ( n ) ) N q ( U ( m ) ) and so the sequence ( N q ( U n )) n ⩾ 1 is not sub-additive for q  = 1. In particular a notion of q -top ological en tropy should b e deﬁned diﬀerently . Inspired b y the previous discussion and Lemma 10.1 , w e deﬁne the q - top olo gic al entr opy of σ b y h top,q ( σ ) = sup U 1 1 − q lim sup n →∞ 1 n N ( U ( n ) ) , where N ( U ( n ) ) denotes the smallest cardinality of an op en sub cov er of U ( n ) . 11. Appendix C - Renyi entropy Deﬁnition 11.1 . The q -R enyi entr opy for p = ( p 1 , p 2 , ..., p d ) is deﬁned as H R q ( p ) = log( P d j =1 p q j ) 1 − q . (11.1) Giv en 0 < q < 1, one can show that H R q ( p ) = F ( H q ( p )) for every probabilit y vector p = ( p 1 , p 2 , ..., p n ), where F stands for the monotone increasing map F : R + → R + giv en b y x 7→ F ( x ) = log(1 + (1 − q ) x ) 1 − q . (11.2) A version of the q -entrop y , when considering probability measures deﬁned on sets that are not ﬁnite is presen ted in (1.28) in [ 69 ] and also in (14) in [ 24 ], but our dynamical deﬁnition of q -entrop y has diﬀerent features when compared with the ones in these tw o references. A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 51 In this wa y , the understanding of prop erties of H q can help in the understanding of the Ren yi entrop y H R q . F or the MaxEnt metho d, Tsallis et al [ 74 ] considered tw o forms of in ternal energy constrain ts. In [ 1 ] a third choice was considered. Our main fo cus is on the pressure problem and not on the MaxEnt metho d. It is also true that X j q j log q ( 1 p j ) − X j q j log q ( 1 q j ) ⩾ 0 . (11.3) and that the maximal v alue of the left-hand side of ( 11.3 ) is equal to d . 12. Appendix D: Basic proper ties of q -exp and q -log functions q -log functions. Under the non-extensiv e point of view it is natural to consider the q -log function (see [ 61 , 71 , 72 ]), deﬁned by u → log q ( u ) = 1 1 − q ( u 1 − q − 1) , (12.1) where q  = 1 and u > 0. W e observe that log q (1) = 0, that log( x ) ⩾ log q ( x ), when q ⩾ 1, that log ( x ) < log q ( x ), when q < 1, and that one reco v ers the classical function log taking the limit of log q as q → 1. See Figure 3 for the graph of log q . Figure 3. Graph of the function log q in case q = 0 . 5 (left) and q = − 0 . 5 (righ t). The domain is (0 , ∞ ). Moreo v er, for each 0 < q < 1 the function u → log q ( u ) is conca ve, log q ( u ) < 0 for ev ery 0 < u < 1, and it satisﬁes log q ( ab ) = log q ( a ) + log q ( b ) + (1 − q ) log q ( a ) log q ( b ) (12.2) and log q  1 p  = − p q − 1 log q ( p ) . (12.3) A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 52 F or 0 < q < 1, 0 < x < 1, the function − log x > 1 1 − q ( x q − 1 − 1); therefore the equilibrium-Boltzman (nondynamical) en tropy h ( p ) = − P j p j log p j satisﬁes h ( p ) ⩾ H q ( p ) . (12.4) In the case q > 1, giv en 0 < x < 1 one has − log x < 1 1 − q ( x q − 1 − 1), hence h ( p ) ⩽ H q ( p ) . (12.5) When q > 0, the largest v alue of the q -entrop y map is attained on the uniformly dis- tributed probabilit y vector p 0 = (1 /n, 1 /n, .., 1 /n ) and it is equal to H q ( p 0 ) = 1 1 − q ( n 1 − q − 1) = log q ( n ) . Example 12.1 ( q -entrop y of Mark o v measures) . A tw o by tw o line sto c hastic matrix P with all p ositive entries P = ( P i,j ) i,j =1 , 2 determines a unique stationary Marko v measure µ on { 1 , 2 } N . As P 11 + P 21 = 1 = P 12 + P 22 , these probabilit y measures µ are indexed b y P 12 , P 2 , 1 ∈ (0 , 1) × (0 , 1). The vector of probabilit y π =  π 1 π 2  is the one such that π P = π . The probability of the cylinder set ij is π i P i,j . F or a stationary Marko v measure µ , the Jacobian J in the cylinder ij is equal to Q ij = P ij π i π j . The classical entrop y of µ is given b y h ( µ ) = − P 2 i,j =1 π i P ij log P ij . In the non-extensiv e case, for q > 0, we get the conca v e function H q ( µ ) = 2 X i,j =1 π i P ij log q ( 1 Q ij ) . Figure 4 b elow sho ws the graph of the v alues of the conca ve function H q ( µ ), as a function of P 12 , P 2 , 1 ∈ (0 , 1) × (0 , 1), when q = 0 . 9. W e p oin t out that for v alues q > 1 the function H q ( µ ) is also concav e. Figure 4. A Marko v stationary probability is determined by the v alues P 12 , P 21 of a line sto c hastic matrix P . Ab o v e the graph of the q -en trop y H q ( µ ), when q = 0 . 9, as a function of ( P 12 , P 21 ). The domain of the concav e function is (0 , 1) × (0 , 1) A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 53 R emark 12.2 . Giv en tw o probability measures p = ( p 1 , p 2 , ..., p n ) , q = ( q 1 , q 2 , ..., n n ), by Jensen inequalit y X j q j log q ( p j q j ) ⩽ log q ( X j q j p j q j ) = log q (1) = 0 . In particular, the probability v ectors p i = (0 , . . . 0 , 1 , 0 , . . . , 0) hav e zero p -en tropy . q -exp functions. F or q > 0, with q  = 1, the in v erse of log q is the q -exp function. This concept is necessary for the deﬁnition of the q -Ruelle op erator. Recall that the q -exp function is deﬁned by u → e u q = exp q ( u ) = (1 + (1 − q ) u ) 1 1 − q , for ev ery q  = 1 and u > 0, that it is conv ex and exp q (0) = 1 and the image of the function exp q is (0 , ∞ ). F or the graph of exp q see Figure 5 . Figure 5. Graph of exp q . On the left q = 0 . 5 and its domain is ( − 2 , ∞ ). On the righ t q = − 0 . 5 and its domain is ( −∞ , 2). Moreo v er, function e x q is the solution of the ordinary diﬀerential equation dy ( x ) dx = y ( x ) q with initial condition y (0) = 1, and exp q ( − log q (1 /x )) ⩽ x (12.6) and it is not the identit y map. R emark 12.3 . The follo wing holds: (1) If q > 1: exp q ( u ) > 0 for every u < 1 q − 1 ; (2) If 0 < q < 1: exp q ( u ) > 0 for every u < 1 q − 1 When q < 1 and q close to 1, we get that exp q ( u ) > 0 for very negative v alues of u . 12.1. Some prop erties. In what follows we list some of the relev an t prop erties of the q -exp onen tial functions, some of whic h used in the text, whic h illustrate the origin of the A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 54 non-additivit y of the non-extensive entrop y function. a ) e z q e − z 2 − q = ( e z q ) q e − q z 1 q = 1 (12.7) b ) e x + y +(1 − q ) x y q = e x q e y q (12.8) c ) e a + b q − e a q = − [1 + a (1 − q )] 1 1 − q + [1 + ( a + b )(1 − q )] 1 1 − q (12.9) d ) e a + b q e a q = [1 + a (1 − q )] − 1 1 − q [1 + ( a + b )(1 − q )] 1 1 − q (12.10) e ) ( e x q ) − 1 − e − x q ⩾ 0 , if x > 1 , and ( e x q ) − 1 − e − x q ⩽ 0 , if x < 1 (12.11) f ) d dx log q ( x ) = 1 x q (12.12) g ) d dx e x q = ( e x q ) q . (12.13) h ) P j = p q j P d k =1 p q k ⇔ p q i = P i ( P d j =1 P 1 /q j ) q (12.14) i ) P j = p j P d k =1 p k ⇔ p i = P i ( P d j =1 P 1 /q j ) q (12.15) j ) ( e x q ) a = e a x 1 − 1 − q a , for any a and q (12.16) k ) log q ( x ) = (1 + (1 − m ) log m ( x )) 1 − q 1 − m 1 − q , for any m and q (12.17) l ) (T a ylor expansion) e x q = 1 + ∞ X n =1 1 n ! Q n − 1 ( q ) x n for an y q > 0 (12.18) where Q n − 1 ( q ) = q (2 q − 1)(3 q − 2) ... [ n q − ( n − 1)] . m ) (T a ylor expansion) log q (1 + x ) = x + ∞ X n =2 ( − 1) n +1 1 n ! Π n − 2 j =0 ( q + j ) x n (12.19) for an y q > 0 n ) (First deriv ativ e) d ( e α + β q − e α q ) dβ = [1 + ( α + β )(1 − q )] − 1+ 1 1 − q , q > 0 (12.20) o ) (Second deriv ativ e) d 2 ( e α + β q − e α q ) d 2 β = ( − 1 + 1 1 − q ) (1 − q ) [1 + ( α + β )(1 − q )] − 2+ 1 1 − q (12.21) for an y q > 0 p ) e a q e b 2 − q = e − 2( − 1+ | b − 2 | | 2+ a | ) 2 − q (12.22) q ) − log q ( 1 e y 2 − q ) = − 1 + (1 + y ( q − 1)) − 1 q − 1 , (12.23) r ) e x + y 2 − q = e − 1+ e y ( q − 1) (1+( q − 1)( x + y )) − 1 1 − q q e y , (12.24) s ) e − log q (1 /y ) 2 − q = y (12.25) t ) d ds e f ( s ) 2 − q = (1 + ( q − 1)( f ( s ))) 2 − q q − 1 ( f ′ ( s )) (12.26) A DYNAMICAL APPR OA CH TO NON-EXTENSIVE THERMOD YNAMICS 55 Ac kno wledgemen ts. W e would like to thank L. Cioletti for providing us with references [ 70 , 71 ], and encouraging us to study this topic. AOL was partially supp orted by CNPq- Brasil. 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E-mail: arturoscar.lop es@gmail.com P aulo V arandas Departamen to de Matem´ atica da Univ ersidade de Av eiro Campus Univ ersit´ ario de Santiago 3810-193 Av eiro, Portugal E-mail: paulo.v arandas@ua.pt

A Dynamical Approach to Non-Extensive Thermodynamics

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