Dominated Splitting and Pesins Entropy Formula

Dominated Splittin g and P esin’s Entrop y F orm ula W enxiang Sun ∗ LMAM, Sc ho ol of Mathematical Sciences, P eking Univ ersit y , Beijing 100871, China E-mail: sun wx@math.pku.edu .cn Xueting Tian † Sc ho ol of Mathematical Sciences, P eking Univ ersit y , Beijing 100871 , C hina E-mail: txt@pku.edu.cn Abstract Let M b e a c ompact manifold and f : M → M b e a C 1 diffeomor- phism on M . If µ is an f -inv ariant p robabilit y measure whic h is abso- lutely con tinuous relativ e to Leb esgue measure and for µ a. e. x ∈ M , there is a dominated splitting T or b ( x ) M = E ⊕ F o n its orbit or b ( x ), then we giv e an estimation th rough L y apuno v c haracteristic exp onen ts from b elo w in P esin’s en trop y formula, i.e., the metric entrop y h µ ( f ) satisfies h µ ( f ) ≥ Z χ ( x ) dµ, where χ ( x ) = P dim F ( x ) i =1 λ i ( x ) and λ 1 ( x ) ≥ λ 2 ( x ) ≥ · · · ≥ λ dim M ( x ) are the Ly apunov exp onen ts at x with r esp ect to µ. Consequentl y , by u s- ing a dichoto my for generic v olume-preserving diffeomorphism we sho w that Pe sin’s e ntrop y formula holds f or generic volume-preserving dif - feomorphisms, whic h generalizes a result of T ahzibi [12] in dimens ion 2. ∗ Sun is suppo r ted b y National Natural Science F oundation ( # 106710 0 6, # 10831 003) and National Basic Resear ch Progra m of China(973 Pr ogram)(# 20 06CB80 5903) † Tian is the cor resp onding author. Key words and phrases: metric entropy , Lyapuno v e x po nent s, Pesin’s entropy formula, dominated splitting AMS Review: 37A05, 37A35 , 37 D2 5, 3 7 D30 1 1 In tro d uction T o estimate metric en tropy through Ly apunov expo nen ts is an imp ortan t topic in differen tial ergo dic theory . In 1977 Ruelle [11] go t from ab o v e an estimate of metric en trop y of a n in v arian t measure, and P esin[10] in 1 9 78 g ot from b elow an estimation o f metric en tr o p y of an in v ariant measure a bsolutely con tin uous relativ e to Leb esgue measure and th us got a so called P esin’ en trop y form ula. P esin’s pro o f is based o n the stable ma nif o ld theorem. In 1980 Ma ˜ n ´ e [7] gav e another inge nious and very simple pro of without using the theory of stable manifolds. In 1985 Ledrappier and Y oung[4] generalized the formula to all SRB measures, no t necessarily a bsolutely contin uous relative to L eb esgue measure. There are also more generalizations[5, 6]. P esin’s en tropy formula b y P esin a nd by Ma ˜ n ´ e a nd by others assumes that not o nly the differen tiabilit y of the giv en dynamics is of class C 1 but also that the first deriv ative satisfies an α -H ¨ o ld er condition for some α > 0. It is interes t- ing to in v estigate P esin’s en t rop y formu la under the w eak er C 1 differen tiabilit y h yp othesis plus some additional condition, for example, dominated splitting. The aim of this pap er is t o pro v e that Pes in’s entrop y formu la remains true for in v aria nt probabilit y measure absolutely contin uous relativ e to Leb esgue measure in the C 1 diffeomorphisms with dominated splitting. In the pro of of [7], the combin ation of the graph tr ansform metho d (Lemma 3 there) and the distortion prop erty deduced from the H ¨ o lder condition of the deriv ative pla y imp ortan t roles. The domination assumption in o ur C 1 diffeomorphism helps us to ov ercome muc h trouble. Our pro o f follows Ma ˜ n´ e without using the theory of stable manifolds, as noted by Katok that it seems that Ma˜ n ´ e’s pro of can also b e extended to the more general fra mew or k. T ahzibi show ed in [12] that there is a residual subset R in C 1 v olume- preserving surface diffeomorphisms suc h that ev ery system in R satisfies P esin’s en tropy f o rm ula. As an consequence our main Theorem 2.2 and a result of Bo c hi and Viana[2], w e generalize the r esult of T ahzibi in to an y di- mensional case. 2 Results Before stating our main results w e need to intro duce the concept of dominated splitting. D enote the minimal norm of a linear map A b y m ( A ) = k A − 1 k − 1 . 2 Definition 2.1. L et f : M → M b e a C 1 diffe omo rphism on a c omp ac t R emainnia n manifold. (1). (Dominate d splitting at one p oint) L et x ∈ M and T or b ( x ) M = E ⊕ F b e a D f − invariant s plitting on or b ( x ) . T or b ( x ) M = E ⊕ F is c a l le d to b e N ( x ) -dominate d at x , if ther e exists a c onstant N ( x ) ∈ Z + such that k D f N ( x ) | E ( f j ( x )) k m ( D f N ( x ) | F ( f j ( x )) ) ≤ 1 2 , ∀ j ∈ Z . (2). (Dominate d spli tting on an invariant set) L et ∆ b e a n f -invariant set and T ∆ M = E ⊕ F b e a D f − invariant splitting on ∆ . We c al l T ∆ M = E ⊕ F to b e a N -dominate d splitting, if ther e exists a c onstant N ∈ Z + such that k D f N | E ( y ) k m ( D f N | F ( y ) ) ≤ 1 2 , ∀ y ∈ ∆ . F or a B or el measurable map f : M → M o n a compact metric space M and an f − in v arian t measure µ , we denote by h µ ( f ) the metric en tr o p y . No w w e state our results as follows. Theorem 2.2. L et f : M → M b e a C 1 diffe omo rphism on a c omp ac t R e- mainnian manifo l d . L e t f pr eserve an in v ariant pr ob ability me a sur e µ which is absolutely c ontinuous r elative to L eb esgue me asur e. F or µ a.e . x ∈ M , denote by λ 1 ( x ) ≥ λ 2 ( x ) ≥ · · · ≥ λ dim M ( x ) the Lyapunov exp onents at x. L et m ( · ) : M → N b e an f -invariant me asur able function. I f for µ a. e. x ∈ M , ther e is a m ( x ) -dominate d splitting: T or b ( x ) M = E or b ( x ) ⊕ F or b ( x ) , then h µ ( f ) ≥ Z χ ( x ) dµ, wher e χ ( x ) = P dim F ( x ) i =1 λ i ( x ) . In p articular, if for µ a. e. x ∈ M , E ( x ) and F ( x ) c oincide with the sum of the Osele de c subbund les c orr esp onding to ne gative Lyapunov exp onen ts and non- ne gative Lyapunov exp onents r esp e ctively (or, E ( x ) c orr es p onds to non-p ositive Lyapunov exp on e nts and F ( x ) c orr esp onds to p ositive Lyapunov exp onents), then h µ ( f ) = Z χ ( x ) dµ = Z X λ i ( x ) ≥ 0 λ i ( x ) dµ. In other wor ds, Pesin ’s entr op y fo rmula hold s . 3 Remark. R ecall t ha t the w ell kno wn Ruelle’s inequality [11 ] h µ ( f ) ≤ Z X λ i ( x ) ≥ 0 λ i ( x ) dµ is v alid for an y inv aria nt measure of f . Thus , if the inv erse inequality hold, the part icular case of Theorem 2.2 is deduced immediately . So the left work w e need to pro v e is the inv erse inequalit y . Since Y ang hav e prov ed in [13] that for an y diffeomorphism f far a w a y from homo clinic tangency and any f -ergo dic measure µ , the sum of the stable, cen t er and unstable bundles in Oseledec splitting is dominated on supp( µ ) , using Theorem 2.2 w e hav e a direct corollar y a s follo ws. Corollary 2.3. L et f ∈ Diff 1 ( M ) far away fr om hom o clinic tangency and let µ b e an f -er go dic pr ob ab ility me a sur e which is abso lutely c ontinuous r elative to L eb esg ue me asur e. Then f satisfies Pesin ’s entr o py formula, i.e . , h µ ( f ) = X λ i > 0 λ i , wher e λ 1 ≥ λ 2 ≥ · · · ≥ λ dim M ar e the Lyapunov exp on ents with r esp e ct to µ. Let m b e the v olume measure and D iff 1 m ( M ) denote the space of v olume- preserving diffeomorphisms. It is known that the stable bundle and unstable bundle of Anoso v diffeomorphism a re alwa ys dominated, and so are the bun- dles b et we en the stable, cen ter and unstable directions in partially h yp erb olic systems . Th us w e ha v e a direct corolla ry as follo ws. Corollary 2.4. L et f ∈ Diff 1 m ( M ) . If f is an Anosov diff e omorphis m (or, a p artial ly hyp erb olic diff e omorphis m wh i c h satisfies that for m a. e. x, the Lya- punov exp o n ents at x in the c entr al bund le ar e either al l non-p ositive o r al l non-ne g a tive ) , then Pes i n ’s entr opy formula holds. In a Baire space, we say a set is residual if it contains a coun table inter- section o f dense op en sets. W e alwa ys call ev ery elemen t in the residual set to b e a generic p oint. It is know n that ev ery C 1+ α v olume-preserving diffeo- morphism satisfies Pes in’s en tropy formula(see [7, 10]) and the set of C 1+ α (or C 2 ) volume-pres erving diffeomorphisms is dense in D iff 1 m ( M ) , so the set of 4 v olume-preserving diffeomorphisms satisfying P esin’s en trop y formula is dense in Diff 1 m ( M ) . Hence, it is natura l to ask whether generic v olume-preserving diffeomorphisms satisfy P esin’s entrop y formula. This problem is not trivial b ecause A. T ahzibi sho wed in [12] that C 1+ α v olume-preserving diffeomor- phisms are not generic in Diff 1 m ( M ) . Here w e use Theorem 2 .2 to deduce this generic prop ert y . Theorem 2.5. Ther e exists a r esi d ual s ubse t R ⊆ Diff 1 m ( M ) such that f o r every f ∈ R , the metric entr op y h µ ( f ) satisfies Pesin ’s entr opy formula, i.e., h µ ( f ) = Z X λ i ( x ) ≥ 0 λ i ( x ) dm, wher e λ 1 ( x ) ≥ λ 2 ( x ) ≥ · · · ≥ λ dim M ( x ) a r e the Lyapunov exp onents of x with r esp e ct to m. Remark. If dim ( M ) = 2 , this result is firstly prov ed in [12]. 3 Pro o f of Theor e m 2.2 Our pro of will b e based on a general lo w er estimate for metric en tropy , whic h mak es it p ossible to av oid the use of partitions. Let g : M → M b e a map, d b e a metric on M and let δ > 0 . If x ∈ M and n ≥ 0 , define Bow en ball B n ( g , δ , x ) = { y ∈ M | d ( g j ( x ) , g j ( y )) ≤ δ, 0 ≤ j ≤ n } . In other w o rds, B n ( g , δ , x ) = n \ j =0 g − j B δ ( g j ( x )) , where B δ ( g j ( x )) denotes the ball cen tered at x with radius δ . If g : M → M is measurable and µ is a measure on M ( no t necessarily g − in v ariant ), define h µ ( g , δ , x ) = lim sup n → + ∞ 1 n [ − log µ ( B n ( g , δ , x ))] . Lemma 3.1. I f g is me as ur able, µ is a g -invariant pr ob ability me asur e on M and ν ≫ µ is another me asur e o n M (not ne c essarily g -i n variant), then h µ ( g ) ≥ sup δ> 0 Z M h ν ( g , δ , x ) d µ. 5 Pro of This lemma is a particular case of the Prop osition in [7], see P .96 or Lemma 13.4 in [8] for details.  Before going into the pro of of Pes in’s formu la we shall pro v e a t echnic al lemma. The reader familia r with the Hadamard g r a ph transform metho d for constructing in v ariant manifo lds will recognize this lemm a one of the steps of that metho d. In the statemen t of the lemma w e shall use the f o llo wing definitions from [7, 8]. Definition 3.2. L et E b e a norme d sp ac e and E = E 1 ⊕ E 2 b e a spl i tting. Define γ ( E 1 , E 2 ) as the supr emum of the norms of the pr oje ction s π i : E → E i i = 1 , 2 , ass o ciate d with the splitting. Mor e over, w e say that a subset G ⊂ E is a ( E 1 , E 2 ) -gr aph if ther e exists an op en U ⊆ E 2 and a C 1 map ψ : U → E 1 satisfying G = { x + ψ ( x ) | x ∈ U } . The n umb er sup { k ψ ( x ) − ψ ( y ) k k x − y k | x 6 = y ∈ U } is c al le d the disp ersion of G . The following lemma ab out graph tr a nsform o n dominated bundles is a generalization to Lemma 3 in Ma ˜ n´ e[7 ] ab out that on h yp erb olic bundles. Ob- serv e that the main p oint of t he pro of of Lemma 3 there is the gap b et w een t w o h yp erb olic bundles and can be replaced b y the g ap of t w o dominated bundles, our pro of of the fo llo wing lemma is a slight c hange o f the pro of of Lemma 3 in Ma ˜ n ´ e[7]. W e giv e a pro of for completeness. Lemma 3.3. Given α > 0 , β > 0 and c > 0 , ther e exi s ts τ > 0 with the fol lowing pr op erty. I f E is a finite-dimen sional norme d sp ac e a nd E = E 1 ⊕ E 2 a splitting with γ ( E 1 , E 2 ) ≤ α , and F is a C 1 emb e dding of a b al l B δ (0) ⊂ E into another B anach sp ac e E ′ satisfying ( i ) . D 0 F is an isomorphism a nd γ (( D 0 F ) E 1 , ( D 0 F ) E 2 ) ≤ α ; ( ii ) . k D 0 F − D x F k ≤ τ for al l x ∈ B δ (0); ( iii ) . k D 0 F | E 1 k m ( D 0 F | E 2 ) ≤ 1 2 ; ( iv ) . m ( D 0 F | E 2 ) ≥ β ; then for every ( E 1 , E 2 ) -gr aph G with disp ersion ≤ c c ontaine d in the b al l B δ (0) , its image F ( G ) is a (( D 0 F ) E 1 , ( D 0 F ) E 2 ) -gr aph with disp ersion ≤ c. Pro of Iden tit y E with E 1 × E 2 and E ′ with ( D 0 F ) E 1 × ( D 0 F ) E 2 . W rite the map F in the form F ( x, y ) = ( Lx + p ( x, y ) , T y + q ( x, y )) , 6 where L = ( D 0 F ) E 1 , T = ( D 0 F ) E 2 . It follows that the partial deriv ativ es of p and q with resp ect to x and y hav e norm ≤ τ α. Let U ⊂ E 2 b e an op en set and ψ : U → E 1 a map whose graph { ( ψ ( v ) , v ) | v ∈ U } is G . Then, F ( G ) = { ( Lψ ( v ) + p ( ψ ( v ) , v ) , T v + q ( ψ ( v ) , v )) | v ∈ U }} . T o study this set define φ : U → ( D 0 F ) E 2 b y φ ( v ) = T v + q ( ψ ( v ) , v )) . If v , w ∈ U, k φ ( v ) − φ ( w ) k ≥ k T ( v − w ) k − k q ( ψ ( v ) , v ) − q ( ψ ( w ) , w ) k . Using the fa ct that the norm o f t he partial deriv atives of q are ≤ τ α a nd h yp othesis (iii) we o btain k φ ( v ) − φ ( w ) k ≥ m ( T ) k v − w k − τ α ( k ψ ( v ) − ψ ( w ) k + k v − w k ) ≥ ( m ( T ) − τ α (1 + c )) k v − w k . Hence, if τ is so small that m ( T ) − τ α (1 + c ) ≥ β − τ α (1 + c ) > 0 , φ is a homeomorphism of U on to φ ( U ) whose inv erse has Lipsc hitz constan t ≤ ( β − τ α ( 1 + c )) − 1 . In particular, φ ( U ) is op en. Now define ˆ ψ : φ ( U ) → ( D 0 F ) E 1 b y ˆ ψ ( v ) = ( Lψ φ − 1 )( v ) + p ( ψ ( φ − 1 ( v )) , φ − 1 ( v )) . Clearly , F ( G ) = { ( ˆ ψ ( x ) , x ) | x ∈ φ ( U ) . } T o calculate the disp ersion of F ( G ), write ˆ ψ = ˜ ψ φ − 1 where ˜ ψ ( w ) = L ˜ ψ ( w ) + p ( ˜ ψ ( w ) , w ) . Then k ˜ ψ ( w ) − ˜ ψ ( v ) k ≤ k L kk ψ ( v ) − ψ ( w ) k + τ α ( k ψ ( v ) − ψ ( w ) k + k v − w k ) ≤ ( c k L k + τ α (1 + c )) k v − w k . 7 Then the dispersion of F ( G ) is less than or equal to c k L k + τ α (1 + c ) /c m ( T ) − τ α (1 + c ) ≤ c 1 2 m ( T ) + τ α (1 + c ) /c m ( T ) − τ α (1 + c ) = c 1 2 + τ α (1 + c ) /cm ( T ) 1 − τ α (1 + c ) /m ( T ) ≤ c 1 2 + τ α (1 + c ) /cβ 1 − τ α (1 + c ) /β . T aking τ small enoug h, the factor of c is < 1 and the lemma is prov ed.  Lemma 3.4. L et g ∈ Diff 1 ( M ) and Λ b e g -invariant subset of M . If ther e is a 1 -domin a te d splitting on Λ : T Λ M = E ⊕ F , then f o r any c > 0 , ther e exists δ > 0 s uch that for every x ∈ Λ and a ny ( E x , F x ) -gr aph G wi th dis- p ersion ≤ c c ontaine d in B owen b al l B n ( x, δ ) ( n ≥ 0) , its im age g n ( G ) is a ( D x g n E x , D x g n F x ) -gr aph with disp ersion ≤ c. Pro of Let β = min x ∈ M m ( D x g ) . Since dominated splitting can b e extended on the closure of Λ and dominated splitting is alw a ys contin uous(see [1]), w e can tak e a finite constan t α = sup x ∈ Λ γ ( E x , F x ) . F or giv en c > 0 and for the ab ov e α , β , tak e τ > 0 satisfying Lemma 3.3. Since D x g is uniformly con tin uous on M , there is δ > 0 suc h that if d ( x, y ) < δ, one has k D x g − D y g k ≤ τ . By applying Lemma 3.3, w e get the following: F act F or a ny y ∈ Λ and eve ry ( E y , F y )-graph H with disp ersion ≤ c contained in the ball B δ ( y ) , it s image g ( H ) is a (( D y g ) E y , ( D y g ) F y )-graph with disp ersion ≤ c. W e prov e Lemma 3.4 by induction. The conclusion is trivial for n = 0 . Assume it holds for some n ≥ 0, that is, we a ssume that if G is a ( E x , F x )- graph with disp ersion ≤ c con tained in Bo w en ball B n ( g , δ, x ) then g n ( G ) is a ( D x g n E x , D x g n F x )-graph with disp ersion ≤ c. No w let G is a ( E x , F x )-graph with disp ersion ≤ c contained in Bo w en ball B n +1 ( g , δ, x ) . Using B n +1 ( g , δ, x ) ⊆ B n ( g , δ, x ) , G is also con tained in B n ( g , δ, x ) . So, by assumption g n ( G ) is a ( D x g n E x , D x g n F x )-graph with disp ersion ≤ c . T ak e y = g n ( x ) ∈ Λ and let H = g n ( G ). Notice that ( D x g n E x , D x g n F x ) = ( E g n x , F g n x ) = ( E y , F y ) 8 and H = g n ( G ) ⊆ g n ( B n ( g , δ, x )) ⊆ B δ ( g n ( x )) = B δ ( y ) . Th us H is a ( E y , F y )-graph with dispersion ≤ c con tained in B δ ( y ). Using the ab ov e F act , w e ha v e g ( H ) is a (( D y g ) E y , ( D y g ) F y )-graph with disp ersion ≤ c. Observ e that g n +1 ( G ) = g ( H ) and (( D x g n +1 ) E x , ( D x g n +1 ) F x ) = (( D y g ) E y , ( D y g ) F y ) , w e get that g n +1 ( G ) is a (( D x g n +1 ) E x , ( D x g n +1 ) F x )-graph with disp ersion ≤ c .  No w w e are ready to prov e P esin’s formula. Pro of of Theorem 2.2 Put Σ j = { x | dim F ( x ) = j } and let S = { j ≥ 0 | µ (Σ j ) > 0 } . If j ∈ S, let µ j b e the measure on M giv en by µ j ( A ) = µ ( A ∩ Σ j ) µ (Σ j ) for all Borel subset A of M . Then µ = X j ∈ S µ (Σ j ) · µ j and th us b y the affine prop ert y of metric en tropy w e hav e h µ ( f ) = X j ∈ S µ (Σ j ) h µ j ( f ) . Th us, all w e hav e to show is t hat h µ j ( f ) ≥ Z χ ( x ) dµ j . 9 This inequalit y obviously holds for j = 0 . Supp o se j > 0 . Not e that µ ≪ Leb implies µ j ≪ Leb for all j ∈ S. Hence, to simplify t he no t ation w e put µ = µ j , Σ = Σ j . Fix an y ε > 0 . T ak e N 0 so large that the set Σ ε = { x ∈ Σ | m ( x ) ≤ N 0 } has µ -measure larger than 1 − ε . Let N = N 0 ! and g = f N , then the splitting T Σ ε M = E ⊕ F satisfies 1 -dominated with respect to g : k D g | E ( x ) k m ( D g | F ( x ) ) ≤ N m ( x ) − 1 Y j =0 k D f m ( x ) | E ( f j m ( x ) x ) k m ( D f m ( x ) | F ( f j m ( x ) x ) ) ≤ ( 1 2 ) N m ( x ) ≤ 1 2 , ∀ x ∈ Σ ε . Note that Σ ε is f - in v ariant and th us g -in v ariant. In what follows , in order to a v oid a cum b ersome and conceptually unnecessary use of co o rdinate c harts, w e shall t r eat M as if it were a Euclidean space. The reader will observ e t ha t all our argumen ts can b e easily formalized b y a completely straigh tforward use of lo cal co ordinates. Since dominated splitting can b e extended on the closure o f Σ ε and domi- nated splitting is a lw a ys con tinu ous(see [1]), w e can take and fix tw o constants c > 0 and a > 0 so small that if x ∈ Σ ε , y ∈ M and d ( x, y ) < a, then for ev ery linear subspace E ⊆ T y M whic h is a ( E ( x ) , F ( x ))- graph with disp ersion < c w e ha v e   log | d etD y g ) | E | − log | det ( D x g ) | F ( x ) |   < ε. Th us | detD y g ) | E | ≥ | det ( D x g ) | F ( x ) | · e − ε . (3.1) By Lemma 3.4, there exists δ ∈ (0 , a ) suc h that for ev ery x ∈ Σ ε and an y ( E x , F x )-graph G with disp ersion ≤ c contained in the ball B n ( g , δ, x ) ( n ≥ 0) , its image g n ( G ) is a (( D x g n ) E x , ( D x g n ) F x )-graph with disp ersion ≤ c. Let ν b e the Leb esgue measure on M . W e give a claim as follo ws: Claim. F o r ev ery x ∈ Σ ε , h ν ( g , δ, x ) ≥ N χ ( x ) − ε. By Lemma 3.1, this prop ert y will imply that h µ ( g ) ≥ Z M h ν ( g , δ, x ) dµ 10 ≥ Z Σ ε h ν ( g , δ, x ) dµ ≥ Z Σ ε ( N χ ( x ) − ε ) dµ = Z M N χ ( x ) dµ − Z M \ Σ ε N χ ( x ) dµ − ε · µ ( Σ ε ) ≥ Z M N χ ( x ) dµ − N · C · dim ( M ) · µ ( M \ Σ ε ) − ε ≥ Z M N χ ( x ) dµ − N · C · dim ( M ) · ε − ε where C = max x ∈ M log k D x f k . Hence, h µ ( f ) = 1 N h µ ( g ) ≥ Z M χ ( x ) dµ − C · dim ( M ) · ε − ε. Since ε is arbitrary this completes the pro of of our theorem. It remains to pro v e the claim. Fix any x ∈ Σ ε . There exists B > 0 satisfying ν ( B n ( g , δ, x )) = B Z E ( x ) ν [( y + F ( x )) ∩ B n ( g , δ, x )] dν ( y ) for a ll n ≥ 0, where ν also denotes the Leb esgue measure in the subspaces E ( x ) and y + F ( x ) , y ∈ E ( x ) . Th us the claim is reduced to sho wing t ha t lim sup n → + ∞ inf y ∈ E ( x ) 1 n [ − log ν (Λ n ( y ))] ≥ N χ ( x ) − ε, (3.2) where Λ n ( y ) = ( y + F ( x )) ∩ B n ( g , δ, x ) . If Λ n ( y ) is not empt y , b y Lemma 3.4 w e ha v e tha t g n (Λ n ( y )) is a ( E ( g n ( x )) , F ( g n ( x )))-graph with disp ersion ≤ c. T ak e D > 0 suc h that D > v ol( G ) (where v ol( · ) denotes v olume) fo r ev ery ( E ( w ) , F ( w ))-graph G with disp ersion ≤ c con tained in B δ ( w ) , w ∈ Σ ε . Observ e that g n (Λ n ( y )) ⊆ g n B n ( g , δ, x ) ⊆ B δ ( g n ( x )) , g n ( x ) ∈ Σ ε , 11 w e ha v e D > v ol ( g n (Λ n ( y ))) = Z Λ n ( y ) | det ( D z g n ) | T z Λ n ( y ) | dν ( z ) . Since g j (Λ n ( y )) ⊆ g j B n ( g , δ, x ) ⊆ B δ ( g j ( x )) ⊆ B a ( g j ( x )) , j = 0 , 1 , 2 , · · · , n, w e ha v e for any z ∈ Λ n ( y ), d ( g j ( z ) , g j ( x )) < a, j = 0 , 1 , 2 , · · · , n. By inequalit y (3.1), w e hav e | det ( D z g n ) | T z Λ n ( y ) | = n − 1 Y j =0 | det ( D g j ( z ) g ) | T g j ( z ) g j Λ n ( y ) | ≥ n − 1 Y j =0  | det ( D g j ( x ) g ) | F ( g j ( x )) | · e − ε  = | det ( D x g n ) | F ( x ) | · e − nε . Hence, 1 n log D ≥ 1 n log Z Λ n ( y ) | det ( D z g n ) | T z Λ n ( y ) | dν ( z ) ≥ 1 n log Z Λ n ( y ) | det ( D x g n ) | F ( x ) | · e − nε dν ( z ) = 1 n log  ν (Λ n ( y )) · | det ( D x g n ) | F ( x ) | · e − nε  = 1 n log ν (Λ ( y )) + 1 n log | d et ( D x g n ) | F ( x ) | − ε. It follo ws that lim n → + ∞ − 1 n log ν (Λ ( y )) ≥ lim n → + ∞ 1 n log | det ( D x g n ) | F ( x ) | − ε. Com bining this inequalit y a nd follo wing equalit y fr o m Oselede c theorem[9] lim n → + ∞ 1 n log | d et ( D x g n ) | F ( x ) | = N χ ( x ) , w e complete the pro of of (3.2). This completes the pro of o f Theorem 2 .2.  12 4 Pro o f of Theor e m 2.5 In this section w e pro v e Theorem 2 .5. Before that we need a result of Bo c hi and Viana[2]. Theorem 4.1. ([2]) Ther e is a r esidual subset R ⊆ Diff 1 m ( M ) such that f o r every f ∈ R a nd for m a. e. x ∈ M , the Osele de c splitting of f is either trivial(i.e., al l Lyapunov exp onents ar e zer o) or dominate d at x . Pro of of T heorem 2.5 Let R ⊆ Diff 1 m ( M ) b e the same as in Theorem 4.1. T ake and fix a diffeomorphism f ∈ R . F or m a. e. x ∈ M , w e can define χ ( x ) = X λ i ( x ) ≥ 0 λ i ( x ) . By Ruelle’s inequalit y[11], we ha v e h m ( f ) ≤ Z χ ( x ) dm. Th us w e only need to prov e that h m ( f ) ≥ Z χ ( x ) dm. Let Σ 0 = { x ∈ M | the Oselede splitting of f is trivial at x } and Σ 1 = { x ∈ M | the Oselede splitting of f is dominated at x } . Without loss of generality , we assume that m (Σ 0 ) > 0 and m (Σ 1 ) > 0. Let m j b e the measure on M given b y m j ( A ) = m ( A ∩ Σ j ) m (Σ j ) ( j = 0 , 1 ) for all Borel subset A of M . Then m 0 (Σ 0 ) = 1 , m 1 (Σ 1 ) = 1 . More precisely , for m 0 a. e. x, the Oseledec splitting is trivial at x and for m 1 a. e. x, the Oseledec splitting is dominated at x . No te that m = m (Σ 0 ) · m 0 + m (Σ 1 ) · m 1 . 13 Th us b y the affine prop erty of metric en tro py w e ha v e h m ( f ) = m (Σ 0 ) · h m 0 ( f ) + m (Σ 1 ) · h m 1 ( f ) . Based on these analysis w e only need to prov e that h m i ( f ) ≥ Z χ ( x ) dm i , i = 0 , 1 . Since the metric en trop y a r e a lw a ys non-negative, obvious ly w e ha v e h m 0 ( f ) ≥ 0 = Z χ ( x ) dm 0 . Note that m 1 are absolutely contin uous relative t o m . By Theorem 2.2, w e g et h m 1 ( f ) ≥ Z χ ( x ) dm 1 . This completes the pro of of Theorem 2.5.  References . 1. Bonatt i, Diaz, Viana, Dynamics b eyond uniform hyp erb olic i ty: a glob al ge ometric and pr ob abilistic p ersp e ctive , Springer-V erlag Berlin Heidel- b erg, 2005, 287-2 9 3. 2. J. Bo c hi, M. Viana, The Lyapunov exp onents of generic volume pr es e rv - ing an d symple ctic systems , Ann. of Math., 1 61, 2005, 1423-1 485. 3. F. L edrappier, J. Str elcyn, A pr o of of the estimation fr om b elow in Pesin ’s entr opy formula , Ergo d. Th. a nd Dynam. Sys., 2 , 1982, 203-219. 4. F. Ledrappier, L. S. Y o ung, The metric entr o py o f diffe omorphi s ms , Ann. of Math., 122, 1985, 509-539 . 5. Pe idong Liu, Pesin ’s entr opy form ula for endomorphis m , Nagoy a Math. J. 150 (1998) 197-20 9. 6. Pe idong Liu, Entr opy f o rm ula of Pesin typ e f o r non-invertible r andom dynamic al systems, Math. Z . 230 (1999) 201-39 . 7. R. 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