Infinitesimal affine geometry of metric spaces endowed with a dilatation structure

Inﬁnitesimal aﬃne geometry of metric spaces endo w ed with a dilatation structure Marius Buliga Institute of Mathematics , Romanian Academ y P .O. BO X 1-764, R O 014700 Bucure ¸ sti, Romania Marius.Buli ga@imar.ro This version: 31.03.2008 Abstract W e study algebraic and geometric prop erties of metric spaces endo w ed with dilata- tion stru ctures, which are emergent during the passage through smaller and smaller scales. In the limit we obtain a generali zation of metric aﬃne geometry , endow ed with a noncommutativ e vector addition operation and with a mod iﬁed version of ratio of three collinear p oints. This is the geome try of normed aﬃne group spaces, a category whic h con tains the ones of homogeneous groups, Carnot groups or contractible groups. In this category group op erations are not fundamental, but derived ob jects, and the generalization of aﬃne geometry is not based on incidence relations. Keyw ords: contractible groups; Carnot groups; dila tation structures; metr ic tangent spaces; aﬃne a lgebra MSC classes: 2 0F65; 20F19 ; 22A10 1 Con ten ts 1 In troductio n 2 2 Aﬃne structure i n terms of dilatations 5 2.1 Aﬃne algebr a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 F o cus on dilatations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Dilatation structures 12 3.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Axioms of dilatation structure s . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Axiom 0: domains and co domains of dila ta tions . . . . . . . . . . . . . . . . . 13 4 Groups with di latations 14 4.1 Conical gr oups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5 4.2 Carnot gr oups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.3 Contractible groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 7 5 Other examples of dilatation s tructures 18 5.1 Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 8 5.2 Dilatation str uctures on the b oundary of the dyadic tree . . . . . . . . . . . . 19 5.3 Sub-riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6 Prop erties of dilatation structures 21 6.1 T ang ent bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.2 T op o lo gical consider a tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 6.3 Diﬀerent iability with resp ect to dilatation s tr uctures . . . . . . . . . . . . . . 23 7 Inﬁnitesimal aﬃne geometry of di latation structures 23 7.1 Aﬃne transfor mations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 7.2 Inﬁnitesimal linear ity o f dilatation structur es . . . . . . . . . . . . . . . . . . 2 4 7.3 Linear stro ng dilatation structures . . . . . . . . . . . . . . . . . . . . . . . . 27 8 Noncommuta tiv e aﬃne geom e try 28 8.1 Inv er se semigroups and Menelaos theorem . . . . . . . . . . . . . . . . . . . . 29 8.2 On the bar ycentric co ndition . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 8.3 The ra tio of three co llinear p oints . . . . . . . . . . . . . . . . . . . . . . . . . 32 1 In tro duction The p oint of view that dilatations can b e taken a s fundamental o b jects which induce a diﬀerential calculus is relatively well kno wn. The idea is simple: in a vector space V deﬁne the dilatation ba sed at x and of co eﬃcient ε > 0 as the function which asso ciates to y the v alue δ x ε y = x + ε ( y − x ) . Then for a function f : V → W b etw een vector spaces V and W we hav e:  δ f ( x ) ε − 1 f δ x ε  ( u ) = f ( x ) + 1 ε [ f ( x + ε ( u − x )) − f ( x )] , 2 th us the directio nal deriv a tive of f a t x , a long u − x app ears as: f ( x ) + D f ( x )( u − x ) = lim ε → 0  δ f ( x ) ε − 1 f δ x ε  ( u ) . Un til recently there was not muc h interest into the g e neralization of such a diﬀerential calculus, based on o ther dila tations than the usual ones, pro bably b ecaus e nob o dy knew any fundamentally diﬀerent example. This changed gradua lly due to diﬀerent lines of resea rch, like the study of hypo elliptic op erator s H¨ or mander [22], harmo nic analysis on homogeneous g r oups F o lland, Stein [14], probability theory on groups Ha zo d [2 0], Sieb ert [27], studies in ge ometric analy sis in met- ric spaces in r elation with sub- riemannian geometry Bella ¨ ıche [2], groups with p oly nomial growth Gr omov [1 8], or Marg ulis type r igidity res ults Pansu [26]. Another line of resea rch c o ncerns the diﬀerential calculus over g eneral base ﬁelds and rings, Ber tr am, Gl¨ ockner and Neeb [3]. A s the author s expla in, it is p ossible to construct such a diﬀerential calculus without using the speciﬁc prop erties of the base ﬁeld (or ring ). In their appro ach it is no t made a distinction be tw een real and ultrametric diﬀerential calculus (and even not b etw een ﬁnite dimensio na l a nd inﬁnite dimensional diﬀerential calc ulus). They point out that diﬀeren tial calculus (int egral calculus not included) seems to b e a part of ana ly sis which is completely general, based o nly on elementary r esults in linea r algebr a and top ology . The diﬀerential calc ulus pro po sed b y Be rtram, Gl¨ ockner a nd Neeb is a g eneralizatio n of “classica l” calculus in to po logical vector spaces over general base ﬁelds, and even ov e r rings. The op eration of vector a dditio n is therefor e a belia n, mo diﬁcatio ns b eing made in relation with the multiplication by scalars. A diﬀeren t idea , emergent in the studies co ncerning g eometric analysis in metr ic spac es, is to establish a diﬀerential ca lculus in homogeneous gro ups, in particular in Ca rnot g r oups. These are nonco mmu tative versions of topo logical v ector space s, in the sense that the oper - ation of addition (of “vectors”) is replaced by a noncommutativ e group op e ration a nd there is a re pla cement of multiplication by scala rs in a general base ﬁeld with a multiplicativ e action of (0 , + ∞ ) by group automorphisms . In fact this is only a part of the nonsmo oth calculus encountered in geometric a nalysis on metric s paces. F or a s urvey see the pa p er b y Heinonen [21]. The o b jects of interest in nonsmo oth calculus as descr ibe d by Heinonen a re s paces of homog eneous t ype, or metric measured spac e s where a genera lization o f Poincar´ e inequality is true. In such spaces the diﬀerential calculus go es a lo ng wa y : Sobo lev spaces, diﬀerentiation theo rems, Hardy spa ces. It is noticeable that in such a g eneral situation we don’t have eno ugh structure to deﬁne diﬀerentials, but only v ar ious constructions cor r esp onding to the no rm of a diﬀere ntial of a function. Nevertheless see the r emark able result of Cheeger [10], who prov es that to a metric measure space sa tisfying a Poincar´ e inequality we can asso ciate an L ∞ cotangent bundle with ﬁnite dimensio nal ﬁb ers . Other imp or tant works which might also be re le v ant in re la tion to this pap er are David, Semmes [12], where space s with arbitra ry small neighbouho o ds containing simila r images of the whole spa ce are studied, or David, Semmes [13], where they study rectiﬁability pr op erties of subsets of R n with arbitrar y small neig hbo urho o ds containing “ big pieces of bi-Lipschitz images” of the whole subset. A pa rticular case of a space o f homog eneous t yp e where more ca n b e said is a normed ho- mogeneous group, deﬁnition 2 .3. According to [1 4] p. 5, a ho mogeneous group is a connected and simply co nnected Lie group who se Lie a lgebra is endow ed with a family o f dilatations { δ ε : ε ∈ (0 , + ∞ ) } , which are algebra a utomorphisms, s imult aneously dia gonaliza ble. As in this case the exp onential of the gr o up is a bijective mapping, we may transform dilatations of 3 the algebra into dila tations of the gr oup, therefore homo g eneous groups are conical gro ups. Also, they ca n b e descr ib ed as nilp otent Lie g roups p ositively graded. Carnot groups a r e homogeneous groups which a re stratiﬁed, meaning that the ﬁrst non- trivial element of the gradua tion genera tes the whole gro up (or alg ebra). The interest into such groups c ome from v ar io us sour ces, rela ted mainly to the study of hypo -elliptic op erato rs H¨ ormander [22], and to extensions of ha rmonic analy sis F olland, Stein [1 4]. Pansu introduced the ﬁr st really new example of such a diﬀer e nt ial calculus base d o n other than usual dilatations: the ones which are a sso ciated to a Carnot gro up. He proved in [26] the potential of what is no w called Pansu deriv ative, b y providing an a lter native pro of of a Marg ulis rig idity t ype result, as a co rollar y of the Ra demacher theorem for Lip- schitz functions on Car not gr oups. Rademacher theorem, stating that a L ips chit z function is deriv able almos t everywhere, is a mathematica l c r ossro ad, b eca use there meet measure theory , diﬀerential calculus and metr ic geometry . In [26] Pansu proves a gener alization of this theorem for his new deriv a tive. The challenge to extend Pansu r esults to g e neral regula r sub-riemannian manifolds, taken by Margulis, Mostow [24] [25], V o do py a nov [28] and others , is diﬃcult b ecause on such general metric spa ce there is no natural underlying alg ebraic structure, as in the ca s e o f Carnot groups, where we hav e the g roup o p eration as a non commutativ e repla cement of the op era tion of addition in vector spaces. On a reg ula r sub-riema nnian manifold we hav e to construct simultaneously several o b- jects: tange nt spa ces to a po int in the sub- riemannian space, an op eration of addition o f “vectors” in the tangent space, and a deriv ative of the type considered b y Pansu. Dedi- cated to the ﬁr st tw o ob jects is a string o f pap ers , either directly related to the sub ject, as Bella ¨ ıc he [2], or g rowing on techniques whic h app eared in the pa p e r dedica ted to g r oups of po lynomial gr owth of Gro mov [18], contin uing in the big pa p er Gr omov [1 9]. In these papers dedicated to sub-riemannian geometry the la ck o f a underlying algebr aic structure w as supplanted by using techniques of diﬀeren tial g eometry . At a closer look , this means that in order to construct the fundamentals of a non standard diﬀerential ca lculus, the authors used the class ic al one. This seems to me compara ble to eﬀorts to study hyperb olic geometry on mo dels , like the Poincar´ e disk, instead o f intrinsically explo re the said geo metry . Dilatation str uctures on metric spaces, introduced in [6], describ e the a pproximate self- similarity pr op erties of a metric space. A dilatation s tructure is a notion related, but mor e general, to gr oups and diﬀerential structures. The basic ob jects of a dilatation str ucture are dilatations (o r contractions). The axioms of a dilatation structure set the rules of interaction be t ween diﬀeren t dilatations. The po int of v ie w of dilatation structures is that dilatations are re a lly fundamental ob jects, no t o nly for deﬁning a notion of der iv ative, but a s well for all algebraic structures that we ma y need. This viewp o int is justiﬁed by the following results o btained in [6], explained in a con- densed and improv ed pre s entation, in the ﬁr s t part o f this pap er. A metric spa c e ( X , d ) which admits a strong dilatatio n str ucture (deﬁnition 3.1) has a metr ic tangent spa c e at any po int x ∈ X (theorem 6.1), and any such metric ta ngent space has an algebraic str ucture of a conical gro up (theorem 6.2). Conical gr oups are generaliza tions of homogeneo us Lie gro ups, but also of p-adic nilp o- ten t gro ups, or of g e ne r al contractible groups. A co nical g roup is a lo cally compact gro up endow e d with a family of dila tations { δ ε : ε ∈ Γ } . Here Γ is a lo cally compact ab elian gro up with an as so ciated mor phism ν : Γ → (0 , + ∞ ) whic h distinguishes an end of Γ, namely the ﬁlter generated by the pre-imag es ν − 1 (0 , r ), r > 0. This end, is denoted by 0 and ε ∈ Γ → 0 means ν ( ε ) → 0 in (0 , + ∞ ). Any contractible g roup is a conical gro up and to a ny co nic a l 4 group we can asso ciate a family of c o ntractible groups. The structur e of co nt ractible gr oups is known in some detail, due to Sieb ert [27], W ang [31], Gl¨ ockner and Willis [1 6], Gl¨ ockner [15] and references therein. By a cla ssical r esult o f Sieber t [27] pr op osition 5.4, we can characterize the a lgebraic structure of the metric tangent spaces asso ciated to dilatation structures of a certain k ind: they are ho mogeneous groups (coro lla ry 6.3). The coro llary 6 .3 thu s represents a genera liza- tion of diﬃcult results in sub-riemannia n geometr y co nc e r ning the structure of the metric tangent s pace at a point o f a r egular sub- riemannian manifold. This line of resea rch is pursued further in the pap er [9]. Morphisms of dilatation structur e s g e neralize the notion of aﬃne transformatio n. A dilatation s tructure on a metr ic space induce a family of dilatation structures o n the same space, at diﬀerent sca les. W e explain that canonical morphis ms be t ween these induced dilatation structure s lead us to a kind of emerg e nt aﬃnity o n smaller a nd smaller scale. Finally w e characterize contractible groups in terms of dilatation s tr uctures. T o a normed contractible gro up we can naturally ass o ciate a linear dilatation s tructure (prop osition 7 .11). Conv ersely , b y theor e m 7.12 a ny linear and s trong dilatation structure comes from the dilatation structure of a normed contractible group. W e ar e thus led to the intro duction of a noncommutativ e aﬃne geometry , in the spir it of Bertram “aﬃne algebr a”, which is co mmut ative accor ding to our p oint of view. In such a ge o metry incidence relations are no longer relev a nt, b eing replaced b y algebraic axioms concerning dilatations. W e deﬁne a version of the ra tio o f thr e e co llinear po int s (replaced by a “r atio function” which a sso ciates to a pair of p oints and tw o p ositive num b er s the third po int) and we prov e that it is the ba sic inv ariant of this g eometry . Moreov er, it turns out that this is the geometry o f normed aﬃne group spa ces, a notion whic h is to c onical groups as a nor med aﬃne space is to a normed top olo gical vector space (theorem 2.5). 2 Aﬃne structure in terms of dilatations 2.1 Aﬃne algebra Bertram [4] Theorem 1 .1 (her e theor em 2.1) and pa ragr aph 5.2 , prop oses the following algebraic description o f aﬃne geometry and of aﬃne metr ic g e ometry ov er a ﬁeld K of characteristic diﬀerent fr o m 2 , which is not based on incidence notions, but on a lgebraic relations co ncerning “pro duct maps”. He then pursues to the developmen t of generalized pro jective geometr ie s and their relations to Jor da n algebras. F or our purp oses, we changed the name of “ pro duct ma ps” (see the theorem b elow) from “ π ” to “ δ ”, mo re precise ly: π r ( x, y ) = δ x r y F urther, in theor e m 2 .1 and deﬁnition 2.2 is explained this p oint of view. Theorem 2.1 The c ate gory of aﬃne sp ac es over a ﬁeld K of char acteristic diﬀer ent fr om 2 is e quivalent with the c ate gory of sets V e quipp e d with a family δ r , r ∈ K , of “pr o duct maps” δ r : V × V → V , ( x, y ) 7→ δ x r y satisfying the fol lowi ng pr op erties (Af1) - (Af4): (Af1) The map r 7→ δ x r is a homomorphism of the unit gr oup K × into the gr oup of bije ctions of V ﬁx ing x , that is δ x 1 y = y , δ x r δ x s y = δ x r s y , δ x r x = x 5 (Af2) F or al l r ∈ K and x ∈ V the map δ x r is an endomorphism of δ s , s ∈ K : δ x r δ y s z = δ δ x r y s δ x r z (Af3) The “ b aryc entric c ondition ”: δ x r y = δ y 1 − r x (Af4) The gr oup gener ate d by the δ x r δ y r − 1 ( r ∈ K × , x, y ∈ V ) is ab elian, that is for al l r , s ∈ K × , x, y , u, v ∈ V δ x r δ y r − 1 δ u s δ v s − 1 = δ u s δ v s − 1 δ x r δ y r − 1 Mor e pr e cisely, in every aﬃne sp ac e over K , the maps δ x r y = (1 − r ) x + ry (2.1.1) with r ∈ K , satisfy (Af1) - (Af4). Conversely, if pr o duct maps with the pr op erties (A f1) - (Af4) ar e given and x ∈ V is an arbitr ary p oint then u + x v := δ x 2 δ u 1 2 v , r u := δ x r u deﬁnes on V the stru ctur e of a ve ctor sp ac e over K with zer o ve ctor x , and this c onstruction is inverse to the pr e c e ding one. Aﬃne maps g : V → V ′ in the u sual sense ar e pr e cisely the homomorph isms of pr o duct maps, t hat is maps g : V → V ′ such that g π r ( x, y ) = π ′ r ( g x, g y ) for al l x, y ∈ V , r ∈ K . W e shall use the name “real normed a ﬃne spac e ” in the following sense. Deﬁnition 2.2 A r e al n orme d aﬃne sp ac e is an aﬃne sp ac e V over R to gether with a distanc e fun ction d : V × V → K such that: (Af5) for al l x ∈ V k · k x := d ( x, · ) : V → K is a norm on the ve ctor sp ac e ( V , x ) with zer o ve ctor x . (Af6) t he distanc e d is tr ans lation invariant: for any x, y , u, v ∈ V we have: d ( x + u v , y + u v ) = d ( x, y ) W e rema rk that the ﬁeld of pro duct maps δ x r (together with the distance function d for the metr ic cas e) is the central ob ject in the constr uction of aﬃne geo metry ov er a g eneral ﬁeld. 2.2 F oc us on dilatations There is ano ther, but related, wa y of generalizing the a ﬃne geometry , which is the o ne of dilatation structures [6]. In this a pproach pro duct maps of Bertra m are r eplaced by “dilatations” . F or this we have to r eplace the ﬁeld K by a commutativ e group Γ (instead o f the mul- tiplicative gro up K × ) endow ed with a “v alua tio n map” ν : Γ → (0 , + ∞ ), which is a g r oup morphism. W e write ε → 0, ε ∈ Γ, for ν ( ε ) → 0 in (0 , + ∞ ). W e keep a xioms like (Af1), (Af2) (from Theorem 2.1), but we mo dify (Af5) (from Deﬁnition 2.2). There will b e one 6 more axiom co ncerning the r elations betw een the distance a nd dilatations. This is explained in theorem 2.5. The conditions app ear ing in theor em 2.5 are a particula r ca s e of the system of axioms of dilata tion s tructures, introduced in [6]. Dilatation structures a r e a lso a gener alization of homog eneous gr o ups, deﬁnition 2 .3, in fact we a r rived to dilatatio n structures after an eﬀort to ﬁnd a common a lgebraic and analytical gr o und for homog eneous groups and sub- riemannian manifolds. The axioms of a dilatation structure are partly algebra ic a nd par tly of a n analy tical nature (by using uniform limits). Metric spaces endowed with dilatation structures hav e bea utiful prop er ties. The most impo rtant is that for any p oint in suc h a spa ce there is a tang e nt space (in the metric sense) realized as a “normed conical group”. Any normed conical group ha s an asso ciated dilatation structure which is “linear” in the sense that it satisﬁes (Af2). How ever, conical gro ups form a family muc h larger than aﬃne spaces (in the usual sense, ov er R or C ). Building blo cks of conical gro ups are homogeneous gr oups (graded Lie g roups) or p-adic versions of them. By r enouncing to (Af3) and (Af4) we thus allow nonco mmut ativity of the “vector addition” op eration. Let us expla in how we can re c ov er the usual aﬃne g eometry fro m the viewp oint of dilatation structures. F or simplicity we tak e here Γ = (0 , + ∞ ) and V is a rea l, ﬁnite dimensional vector space. Here is the deﬁnition of a no rmed ho mo geneous gro up. See sectio n 4.2 for mo re de ta ils on the par ticular case o f s tr atiﬁed homo geneous groups. Deﬁnition 2.3 A norme d homo gene ous gr oup is a c onne cte d and simply c onne cte d Lie gr oup whose Lie algebr a is endowe d with a family of dilatatio ns { δ ε : ε ∈ (0 , + ∞ ) } , which ar e algebr a au t omorphisms, simultane ously diagonalizable, to gether with a homo gene ous norm. Sinc e the Lie gr oup exp onent ial is a bije ction we shal l identify the Lie algebr a with the Lie gr oup, thu s a norme d homo gene ous gr oup is a gr oup op er ation on a ﬁnite dimensional ve ctor sp ac e V . The op er ation wil l b e denote d mult iplic atively, with 0 as neutr al element, as in F ol land, S tein [14]. We thus have a line ar action δ : (0 , + ∞ ) → Lin ( V , V ) on V , and a homo gene ous norm k · k : V → [0 , + ∞ ) , s uch t hat: (a) for any ε ∈ (0 , + ∞ ) the tra nsformation δ ε is an automorphism of the gr oup op er ation: for any x, y ∈ V we have δ ε ( x · y ) = δ ε x · δ ε y (b) the family { δ ε : ε ∈ (0 , + ∞ ) } is simultane ously diagonalizable : ther e is a ﬁnite dir e ct sum de c omp osition of the ve ctor sp ac e V V = V 1 + ... + V m such that for any ε ∈ (0 , + ∞ ) we have: x = m X i =1 x i ∈ V m 7→ δ ε x = m X i =1 ε i x i . (c) the homo gene ous norm has the pr op erties: (c1) k x k = 0 if and only if x = 0 , (c2) k x · y k ≤ k x k + k y k for any x, y ∈ V , (c3) for any x ∈ V and ε > 0 we have k δ ε x k = ε k x k 7 Deﬁnition 2.4 T o a norme d homo gene ous gr oup ( V , δ, · , k · k ) we asso ciate a norme d aﬃne gr oup sp ac e ( V , + · , δ · · , d ) . Her e we use the sign “ + ” for an op er ation which was denote d multiplic atively, for c omp atibility with the pr evious appr o ach of Bertr am , se e the or em 2.1. The norme d aﬃne gr oup sp ac e ( V , + · , δ · · , d ) is describ e d by the fol lowing p oints: - for any u ∈ V the function + u : V × V → V , x + u v = x · u − 1 · v is the left tr anslatio n of the gr oup op er ation · with the zer o element u . I n p articular we have x + 0 y = x · y . - for any x, y ∈ V and ε ∈ (0 , + ∞ ) we deﬁne δ x ε y = x · δ ε ( x − 1 · y ) and r emark that the deﬁnition is invariant with the choic e of t he b ase p oint for the op er ation in the sense: for any u ∈ V we have: δ x ε y = x + u δ u ε ( inv u ( x ) + u y ) wher e inv u ( x ) is the inverse of x with re sp e ct to t he op er ation + u , (by c omputation we get inv u ( x ) = u · x − 1 · u ), - the distanc e d is deﬁne d as: for any x, y ∈ V we have d ( x, y ) = k x − 1 · y k . A s pr eviously, r emark that the deﬁnition do es not dep en d on t he choic e of the b ase p oint for the op er ation, that is: for any u ∈ V we have d ( x, y ) = k in v u ( x ) + u y k u , k x k u := k u − 1 · x k Equal ly, this is a c onse quenc e of t he invarianc e of t he norm with r esp e ct to left t r ans- lations (by any gr oup op er ation + u , u ∈ V ). Theorem 2.5 The c ate gory of norme d aﬃne gr oup sp ac es is e quivalent with the c ate gory of lo c al ly c omp act metric sp ac es ( X , d ) e quipp e d with a family δ ε , ε ∈ (0 , + ∞ ) , of dilatations δ ε : X × X → X , ( x , y ) 7→ δ x ε y satisfying the fol lowi ng pr op erties: (Af1’) The map ε 7→ δ ε x is a homomorph ism of the multiplic ative gr oup (0 , + ∞ ) into the gr oup of c ontinuous, with c ontinuous inverse functions of X ﬁxing x , that is δ x 1 y = y , δ x r δ x s y = δ x r s y , δ x r x = x (A2) the function δ : (0 , + ∞ ) × X × X → X deﬁne d by δ ( ε, x, y ) = δ x ε y is c ontinu ous. Mor e over, it c an b e c ontinuously extende d to [0 , + ∞ ) × X × X by δ (0 , x, y ) = x and the limit lim ε → 0 δ x ε y = x is uniform with r esp e ct t o x, y in c omp act set. (A3’) for any x ∈ X and for any u, v ∈ X , ε ∈ (0 , + ∞ ) we have 1 ε d ( δ x ε u, δ x ε v ) = d ( u, v ) 8 (A4) for any x, u, v ∈ X , ε ∈ (0 , + ∞ ) let us deﬁne ∆ x ε ( u, v ) = δ δ x ε u ε − 1 δ x ε v . Then we have the limit lim ε → 0 ∆ x ε ( u, v ) = ∆ x ( u, v ) uniformly with re sp e ct to x, u, v in c omp act s et . (Af2’) F or al l ε ∈ (0 , + ∞ ) and x ∈ X the map δ x ε is an endomorphism of δ s , s ∈ (0 , + ∞ ) : δ x r δ y s z = δ δ x r y s δ x r z Mor e pr e cisely, in every norme d aﬃne gr oup s p ac e , the maps δ x ε and distanc e d satisfy (Af1’), (A2), (A3’), (A4), (Af2’). Conversely, if dilatations δ x ε and distanc e d ar e given, such that they satisfy the c ol le ction (Af1’), (A2), (A3’), (A4), (Af2’), for an arbitr ary p oint x ∈ V the fol lowing ex pr ession Σ x ( u, v ) := lim ε → 0 δ x ε − 1 δ δ x ε u ε v to gether with δ x ε and distanc e d deﬁnes on V the structur e of a n orme d aﬃne gr oup sp ac e, and this c onstruction is inverse to the pr e c e ding one. The arr ows of this c ate gory ar e bilipschi tz invertible homomorphisms of dilatations, that is maps g : V → ˆ V such t hat g δ x ε y = ˆ δ gx r g y for al l x, y ∈ V , ε ∈ (0 , + ∞ ) . Mor e over, the c ate gory of r e al norme d aﬃne sp ac es is a sub c ate gory of the pr evious one, namely t he c ate gory of lo c al ly c omp act metric sp ac es ( X , d ) e qu ipp e d with a family δ ε , ε ∈ (0 , + ∞ ) , of dilatations satisfying (Af1’), (A2), (A 3’), (A4), ( A f2’) and (Af3) the “b aryc entric c ondition ”: for al l ε ∈ (0 , 1 ) δ x ε y = δ y 1 − ε x The arr ows of this c ate gory ar e exactly the aﬃne, invertible maps. Pro of. Here we shall prove the ea sy implication, namely wh y the conditio ns (Af1’), (A2), (A3’), (A4), (Af2’) and (Af3) are s a tisﬁed in a real nor med aﬃne space. F or the real no rmed aﬃne space space V let us ﬁx for simplicity a p oint 0 ∈ V and work with the vector space V with zero vector 0. Since a rea l normed aﬃne space is a particular exa mple of a ho mogeneous gr oup, deﬁnition 2.3 and obs e rv atio ns inside apply . The dilatation base d at x ∈ V , of co eﬃcient ε > 0, is the function δ x ε : V → V , δ x ε y = x + ε ( − x + y ) . F or ﬁxed x the dila tations ba sed a t x form a one para meter gro up which contracts any bo unded neigh bo urho o d o f x to a po int, uniformly with resp ect to x . Thus (Af1’), (A2) are satisﬁed. (A3’) is also o bvious. The meaning of (A4) is that using dilatations we can recov er the op eration of addition a nd m ultiplication by sca lars. W e s hall explain this in detail since this will help the understanding of the axio ms of dilatation s tructures, descr ib ed in sectio n 3. F or x, u, v ∈ V and ε > 0 we deﬁne the following comp ositions of dilatations: ∆ x ε ( u, v ) = δ δ x ε u ε − 1 δ x ε v , (2.2.2) Σ x ε ( u, v ) = δ x ε − 1 δ δ x ε u ε ( v ) , inv x ε ( u ) = δ δ x ε u ε − 1 x . 9 The meaning of this functions b ecomes cle a r if we compute: ∆ x ε ( u, v ) = x + ε ( − x + u ) + ( − u + v ) , Σ x ε ( u, v ) = u + ε ( − u + x ) + ( − x + v ) , inv x ε ( u ) == x + ε ( − x + u ) + ( − u + x ) . As ε → 0 we hav e the limits: lim ε → 0 ∆ x ε ( u, v ) = ∆ x ( u, v ) = x + ( − u + v ) , lim ε → 0 Σ x ε ( u, v ) = Σ x ( u, v ) = u + ( − x + v ) , lim ε → 0 inv x ε ( u ) = inv x ( u ) = x − u + x , uniform with resp ect to x, u , v in b ounded sets . T he function Σ x ( · , · ) is a gr oup op er ation, namely the a ddition op eration translated such that the neutral element is x : Σ x ( u, v ) = u + x v . The function inv x ( · ) is the inv erse function with r esp ect to the op era tion + x inv x ( u ) + x u = u + x inv x ( u ) = x and ∆ x ( · , · ) is the diﬀerenc e function ∆ x ( u, v ) = inv x ( u ) + x v Notice that fo r ﬁxed x, ε the function Σ x ε ( · , · ) is not a g roup op eration, ﬁrst of all beca use it is not asso cia tive. Nev ertheless, this function satisﬁes a “shifted” asso ciativit y prop erty , namely Σ x ε (Σ x ε ( u, v ) , w ) = Σ x ε ( u, Σ δ x ε u ε ( v , w )) . Also, the inv erse function inv x ε is not involutiv e, but shifted involutiv e: inv δ x ε u ε ( inv x ε u ) = u . Aﬃne contin uous transfor mations A : V → V admit the following des c r iption in terms of dila ta tions. (W e could dis pe ns e o f contin uit y hypothesis in this situation, but we wan t to illustrate a g eneral p oint of view, descr ib ed further in the pap er). Prop ositi on 2. 6 A c ontinuous tr ansformation A : V → V is aﬃne if and only if for any ε ∈ (0 , 1) , x, y ∈ V we have Aδ x ε y = δ Ax ε Ay . (2.2.3 ) The pro of is a stra ig htf orward consequence of repr esentation formulæ (2.2.2) for the addition, diﬀerence and inv erse oper ations in terms of dilata tions. In par ticular any dila tation is a n aﬃne tra nsformation, hence for any x, y ∈ V and ε, µ > 0 we have δ y µ δ x ε = δ δ y µ x ε δ y µ . (2.2.4) 10 Thu s we recover (Af2’) (see also condition (Af2)). The barycentric condition (Af3) is a consequence of the commutativit y of the addition o f vectors. T he eas y part of the theor em 2.5 is there fo re prov en. The sec ond, diﬃcult part of the theorem is to prov e that axioms (Af1’), (A2), (A3’), (A4), (Af2’) describ e normed aﬃne gr oup spa ces. This is a direct consequence of several general r esults from this pap er: theorem 4.3 and pro p osition 7.11 show that no r med aﬃne group spac es satisfy the axioms, corollar y 6 .3, theorem 7 .12, prop os ition 8.9 and theorem 8.14 show that conversely a space wher e the ax io ms ar e sa tisﬁed is a normed a ﬃne group space, moreov er that in the presence of the baryce nt ric condition (Af3) we g et rea l normed aﬃne spaces.  Some comp ositions of dilata tions a re dilatations. This is prec isely stated in the next theorem, which is equiv a lent with the Menelao s theorem in euclidean ge ometry . Theorem 2.7 F or any x, y ∈ V and ε, µ > 0 such that εµ 6 = 1 ther e exists an un ique w ∈ V such that δ y µ δ x ε = δ w εµ . F or the pr o of see Artin [1]. A straightforward consequence of this theorem is the following result. Corollary 2.8 The inverse semigr oup gener ate d by dilatations of the sp ac e V is made of al l dilatations and al l tr anslations in V . Pro of. Indeed, by theorem 2.7 a compo sition of tw o dila tations with coeﬃcients ε, µ with εµ 6 = 1 is a dilatation. By direct computation, if εµ = 1 then w e obtain transla tio ns. This is in fact compatible with (2.2.2), but is a str onger statemen t, due to the fact that dilatations are aﬃne in the sense o f rela tio n (2.2.4). An y compo s ition b etw een a tra ns lation and a dilatation is again a dilatation. The pro of is done.  The corolla ry 2 .8 allows us to descr ib e the ratio of three collinear points in a w a y whic h will b e g e ne r alized to normed aﬃne group spaces. Indeed, in a re al normed aﬃne space V , for any x, y ∈ V and α, β ∈ (0 , + ∞ ) such that αβ 6 = 1, ther e is an unique z ∈ V and γ = 1 /αβ such that δ x α δ y β δ z γ = id W e easily ﬁnd that x, y , z are collinear z = 1 − α 1 − αβ x + α (1 − β ) 1 − αβ y (2.2.5) the ratio of these three p oints, named r ( x α , y β , z γ ) is: r ( x α , y β , z γ ) = α 1 − αβ Conv ersely , let x, y , z ∈ V which a re collinear , such tha t z is in be t ween x and y . Then we can easily ﬁnd (non unique) α, β , γ ∈ (0 , + ∞ ) such that αβ γ = 1 and δ x α δ y β δ z γ = id . It is then a lmost straig htf orward to prove the well known fac t that a ny aﬃne tr ansfor- mation is also geometrically aﬃne, in the sense that it tra nsforms triples of collinear p oints int o tr iples of collinear points (use commutation with dilatatio ns) and it prese rves the ratio of collinear p oints. (The co nv er se is also tr ue ). 11 3 Dilatatio n structures A dilatation structure ( X , d, δ ) ov er a metric spa ce ( X , d ) is an assig nment to an y p oint x ∈ X of a group of ”dilatations” { δ x ε : ε ∈ Γ } , together with some compatibility conditions betw een the distance and the dilatations a nd b etw een dilatations based in diﬀerent p oints. A basic diﬃculty in stating the axioms o f a dilatation structure is related to the do ma in of deﬁnition and the image of a dilatation. In this subsection w e shall neglect the problems raised by domains and co do mains of dilatatio ns. The a x ioms state that some combinations betw een dilatations and the distance con verge uniformly , with resp ect to some ﬁnite families of p oints in a n arbitrar y compact subset o f the metric spa ce ( X , d ), a s ν ( ε ) conv erges to 0. W e pre s ent here an introduction in to the sub ject o f dilatation structur e s. F or mor e details see Bulig a [6]. 3.1 Notations Let Γ b e a top olo g ical sepa r ated commutativ e group endowed with a contin uous group morphism ν : Γ → (0 , + ∞ ) with inf ν (Γ) = 0. Here (0 , + ∞ ) is taken as a group with m ultiplication. The neutral element of Γ is denoted by 1. W e use the multiplicativ e notation for the o p e ration in Γ. The morphism ν deﬁnes a n inv ariant top ologica l ﬁlter on Γ (eq uiv alently , a n end). In- deed, this is the ﬁlter gener ated by the op en s ets ν − 1 (0 , a ), a > 0. F r om now on we shall name this top ologica l ﬁlter (end) by ”0” and we shall write ε ∈ Γ → 0 for ν ( ε ) ∈ (0 , + ∞ ) → 0. The set Γ 1 = ν − 1 (0 , 1] is a semigroup. W e note ¯ Γ 1 = Γ 1 ∪ { 0 } On the s et ¯ Γ = Γ ∪ { 0 } we ex tend the op era tion on Γ by adding the rules 00 = 0 and ε 0 = 0 fo r any ε ∈ Γ. This is in agreement with the inv ariance of the end 0 with resp ect to tra nslations in Γ. The space ( X , d ) is a complete, lo cally compact metric space. F o r any r > 0 and any x ∈ X we denote by B ( x, r ) the op en ball of center x and radius r in the metric s pa ce X . On the metric space ( X , d ) we w ork with the top ology (and uniformity) induced by the distance. F or any x ∈ X we denote by V ( x ) the to p o logical ﬁlter of o p e n ne ig hbourho o ds of x . The dilatation structures, which will b e introduced so o n, are inv ariant to the op eration of m ultiplication of the distance by a pos itive constant. T he y should als o b e seen, as examples show, a s lo cal ob jects, therefore we may safely supp ose, without r estricting the gene r ality , that all clo sed balls o f r adius at most 5 a r e compact. 3.2 Axioms of dilatation structures W e shall list the ax ioms of a dilatation s tr ucture ( X , d, δ ), in a simpliﬁed form, without concerning ab out domains and codo ma ins of functions. In the next subsection w e shall add the supplementary conditions concerning domains and co domains of dilatations. A1. F or any p oint x ∈ X ther e is an action δ x : Γ → E nd ( X , d, x ) , wher e E nd ( X, d, x ) is the c ol le ction of al l c ontinu ous, with c ontinuous inverse tr ansforma tions φ : ( X, d ) → ( X, d ) such that φ ( x ) = x . This axio m (the s a me as (A1) fro m theorem 2.1 o r theo rem 2.5) tells us that δ x ε x = x for an y x ∈ X , ε ∈ Γ, also δ x 1 y = y for any x, y ∈ X , and δ x ε δ x µ y = δ x εµ y for any x, y ∈ X and ε, µ ∈ Γ. 12 A2. The function δ : Γ × X × X → X deﬁne d by δ ( ε, x, y ) = δ x ε y is c ontinuous. Mor e over, it c an b e c ontinuously extende d to ¯ Γ × X × X by δ (0 , x, y ) = x and the limit lim ε → 0 δ x ε y = x is uniform with r esp e ct t o x, y in c omp act set. W e may alternatively put that the previous limit is unifor m with res pe ct to d ( x, y ). A3. Ther e is A > 1 su ch that for any x ther e exists a function ( u, v ) 7→ d x ( u, v ) , deﬁne d for any u , v in the close d b al l (in distanc e d) ¯ B ( x, A ) , such that lim ε → 0 sup  | 1 ε d ( δ x ε u, δ x ε v ) − d x ( u, v ) | : u , v ∈ ¯ B d ( x, A )  = 0 uniformly with re sp e ct to x in c omp act set. It is easy to see that: (a) The function d x is contin uous as an unifor m limit of contin uous functions on a co mpact set, (b) d x is symmetric d x ( u, v ) = d x ( v , u ) for any u, v ∈ ¯ B ( x, A ), (c) d x satisﬁes the triangle inequality , but it can be a degenerated dis tance function: there might exist v , w such that d x ( v , w ) = 0. W e make the following no tation which generalizes the notation from (2 .2.2): ∆ x ε ( u, v ) = δ δ x ε u ε − 1 δ x ε v . The next ax iom can now be stated: A4. W e hav e the limit lim ε → 0 ∆ x ε ( u, v ) = ∆ x ( u, v ) uniformly with resp ect to x, u, v in c ompact set. Deﬁnition 3.1 A t riple ( X , d, δ ) which satisﬁes A1, A2, A3, but d x is de gener ate for some x ∈ X , is c al le d de gener ate dilatation stru ctur e. If t he triple ( X , d, δ ) satisﬁes A1, A2, A3 and d x is n on-de gener ate for any x ∈ X , then we c al l it a dilatation st ructur e. If a dilatation struct ur e satisﬁes A4 t hen we c al l it str ong dilatation st ructur e. 3.3 Axiom 0 : domains and co domains of dilatations The proble m of doma ins and co domains of dilatation c a nnot b e neglected. In the section dedicated to examples of dilatatio n structures we prese nt the particular cas e of an ultra metric space which is a lso a ball of radius one. As dilatations approximately contract distances , it follows that the co domain of a dilatation δ x ε with ν ( ε ) < 1 ca n not b e the whole space. There are other e x amples showing that we can no t always take the domain of a dila tation to b e the whole s pace. That is because the topolo gy of small balls ca n b e diﬀerent fro m the top ology of big ones (like in the case of a sphere ). 13 F or all these r easons we need to imp os e so me minimal conditions o n the domains and co domains of dila tations. T he s e conditions will b e explained in the following. They will b e considered as par t of a new ax iom, called Axiom 0 . F or any x ∈ X there is an op en neighbourho od U ( x ) of x such that for any ε ∈ Γ 1 the dilatations ar e functions δ x ε : U ( x ) → V ε ( x ) . The sets V ε ( x ) are o p en ne ig hbourho o ds of x . There is a num ber 1 < A such that for a ny x ∈ X w e hav e ¯ B d ( x, A ) ⊂ U ( x ). There is a nu m be r B > A such tha t for any ε ∈ Γ with ν ( ε ) ∈ (1 , + ∞ ) the asso cia ted dila tation is a function δ x ε : W ε ( x ) → B d ( x, B ) . W e hav e the following string of inclusio ns, for any ε ∈ Γ 1 , and a ny x ∈ X : B d ( x, ν ( ε )) ⊂ δ x ε B d ( x, A ) ⊂ V ε ( x ) ⊂ W ε − 1 ( x ) ⊂ δ x ε B d ( x, B ) . In relation w ith the axio m A4 we need the following condition on the c o -domains V ε ( x ): for a ny compact set K ⊂ X there a re R = R ( K ) > 0 and ε 0 = ε ( K ) ∈ (0 , 1) such tha t fo r all u, v ∈ ¯ B d ( x, R ) a nd all ε ∈ Γ, ν ( ε ) ∈ (0 , ε 0 ), we ha ve δ x ε v ∈ W ε − 1 ( δ x ε u ) . These co nditions ar e imp ortant for describing dilatatio n structures on the b oundary of the dyadic tree, for example. In the ﬁrst formulation of the axioms given in [6] some of these assumptions ar e par t of the Axiom 0, others can b e found in the initial for mulation of the Axioms 1, 2, 3 . 4 Groups with dilata tions F or a dila ta tion structur e the metric tangent space s have a gro up structure which is com- patible with dilatations . This structur e, of a normed gro up with dila tations, is int eresting by itself. The notion ha s b een intro duced in [5 ], [6]; we describ e it further. W e shall work further with lo ca l gr oups. Such o b jects are not g roups: they ar e spa c es endow e d with an op eratio n deﬁned only lo cally , sa tisfying the conditions of a uniform g roup. In [5] we use a s lightly non standard deﬁnition of such ob jects. F or the purp oses of this pap er it seems enough to men tion that neighbourho o ds of the neutral element in a uniform group are lo cal groups. See sectio n 3.3 [6] for details ab o ut the deﬁnitio n of lo ca l gr oups. Deﬁnition 4.1 A gr oup with dilatatio ns ( G, δ ) is a lo c al gr oup G with a lo c al action of Γ (denote d by δ ), on G su ch that H0. the limit lim ε → 0 δ ε x = e exists and is uniform with r esp e ct to x in a c omp act neighb ourho o d of the identity e . H1. the limit β ( x, y ) = lim ε → 0 δ − 1 ε (( δ ε x )( δ ε y )) is wel l deﬁne d in a c omp act neighb our ho o d of e and the limit is uniform. 14 H2. the fol lo wing r elatio n holds lim ε → 0 δ − 1 ε  ( δ ε x ) − 1  = x − 1 wher e the limit fr om t he left hand side ex ist s in a neighb ourho o d of e and is un iform with r esp e ct to x . Deﬁnition 4.2 A norme d gr oup with dilatations ( G, δ, k · k ) is a gr oup with dila tations ( G, δ ) endowe d with a c ontinuous norm fun ct ion k · k : G → R which satisﬁes (lo c al ly, in a neighb ourho o d of t he neutr al element e ) the pr op ert ies: (a) for any x we have k x k ≥ 0 ; if k x k = 0 then x = e , (b) for any x, y we have k xy k ≤ k x k + k y k , (c) for any x we have k x − 1 k = k x k , (d) the limit lim ε → 0 1 ν ( ε ) k δ ε x k = k x k N exists, is uniform with r esp e ct to x in c omp act set, (e) if k x k N = 0 then x = e . In a no rmed gro up w ith dilatations we have a na tural left inv ariant distance g iven b y d ( x, y ) = k x − 1 y k . (4.0.1) An y normed group with dilatations has an as so ciated dilatation structure on it. In a g roup with dilatations ( G, δ ) we deﬁne dilatations based in any point x ∈ G b y δ x ε u = xδ ε ( x − 1 u ) . (4.0.2) The following result is theo rem 15 [6]. Theorem 4.3 L et ( G, δ, k · k ) b e a lo c al ly c omp act norme d lo c al gr oup with dilatations. Then ( G, d, δ ) is a dilatation st r u ctur e, wher e δ ar e the dilatations deﬁne d by (4.0.2) and the distanc e d is induc e d by t he norm as in (4.0.1). 4.1 Conical groups Deﬁnition 4.4 A norme d c onic al gr oup N is a norme d gr oup with dilatations such that for any ε ∈ Γ the dilatatio n δ ε is a gr oup morphism and su ch that for any ε > 0 k δ ε x k = ν ( ε ) k x k . A conical gro up is the inﬁnitesimal version of a gro up with dilatations ([6 ] prop osition 2). Prop ositi on 4. 5 Un der the hyp otheses H0, H1, H2 ( G, β , δ, k · k N ) is a lo c al norme d c onic al gr oup, with op er ation β , dilatations δ and homo gene ous n orm k · k N . 15 4.2 Carnot groups Carnot g roups a pp e ar in sub- riemannian g eometry a s mode ls of tang ent spaces, [2], [1 8], [26]. In particular such g roups can b e endow ed with a structure o f sub-r ie mannian manifold. Deﬁnition 4.6 A Carnot (or stra tiﬁe d homo gene ous) gr oup is a p air ( N , V 1 ) c onsisting of a r e al c onne cte d simply c onne cte d gr oup N with a distinguishe d su bsp ac e V 1 of the Lie algebr a Lie ( N ) , such that t he fol lowing dir e ct sum de c omp osition o c curs: n = m X i =1 V i , V i +1 = [ V 1 , V i ] The numb er m is the step of t he gr oup. The numb er Q = m X i =1 i dimV i is c al le d the homo gene ous dimension of the gro up. Because the gr oup is nilp o tent and simply connected, the exp o nential mapping is a diﬀeomorphism. W e s hall ide ntify the gr oup with the algebra, if is not lo cally otherwise stated. The structure that we obta in is a set N endow ed with a Lie brack et and a g roup mu l- tiplication op eratio n, r elated by the B a ker-Campbell- Hausdorﬀ formula. Remark that the group op eratio n is p olynomial. An y Carnot g roup admits a one-par ameter family of dilatatio ns. F or a ny ε > 0, the asso ciated dilatation is: x = m X i =1 x i 7→ δ ε x = m X i =1 ε i x i An y such dilatation is a g roup morphis m and a Lie algebr a mor phism. In a Ca rnot group N let us choo se an euclidean nor m k · k on V 1 . W e shall e ndow the g roup N with a structur e of a sub-r ie ma nnian manifo ld. F or this take the distribution obtained from left translates of the space V 1 . The metric on that distribution is o btained by left translatio n of the inner pro duct r estricted to V 1 . Because V 1 generates (the a lgebra) N then any element x ∈ N can b e written as a pro duct of elements fr om V 1 , in a controlled wa y , desc r ib ed in the fo llowing useful lemma (slight r eformulation o f Lemma 1 .40, F olla nd, Stein [14]). Lemma 4.7 L et N b e a Carnot gr oup and X 1 , ..., X p an orthonormal b asis for V 1 . Th en ther e is a a natura l n umb er M and a function g : { 1 , ..., M } → { 1 , ..., p } such that any element x ∈ N c an b e written as: x = M Y i =1 exp( t i X g ( i ) ) (4.2.3) Mor e over, if x is suﬃciently close (in Euclide an norm) to 0 t hen e ach t i c an b e chosen such that | t i |≤ C k x k 1 /m As a co nsequence we get: Corollary 4.8 The Carnot-Car ath´ eo dory distanc e d ( x, y ) = inf  Z 1 0 k c − 1 ˙ c k d t : c (0 ) = x, c (1) = y , 16 c − 1 ( t ) ˙ c ( t ) ∈ V 1 for a.e. t ∈ [0 , 1]  is ﬁnite for any two x, y ∈ N . The distanc e is obviously left invariant, thus it induc es a norm on N . The Car not-Cara th´ eo dory distance induces a homogeneous no rm on the Carnot g r oup N by the formula: k x k = d (0 , x ). F ro m the inv aria nc e of the distance with re sp ect to left translations we get: for a ny x, y ∈ N k x − 1 y k = d ( x, y ) F or any x ∈ N and ε > 0 we deﬁne the dilata tio n δ x ε y = xδ ε ( x − 1 y ). Then ( N , d, δ ) is a dilatation structure , according to theo r em 4.3. 4.3 Con tractible groups Deﬁnition 4.9 A c ontr actible gr oup is a p air ( G, α ) , wher e G is a top olo gic al gr oup with neutr al element denote d by e , and α ∈ Aut ( G ) is an automorphism of G su ch that: - α is c ontinu ous, with c ontinuous inverse, - for any x ∈ G we have the limit lim n →∞ α n ( x ) = e . F or a contractible gr oup ( G, α ), the automorphism α is compactly contractive (Lemma 1.4 (iv) [2 7]), that is: for each compact s e t K ⊂ G a nd op en set U ⊂ G , with e ∈ U , there is N ( K, U ) ∈ N such that for any x ∈ K a nd n ∈ N , n ≥ N ( K, U ), we ha ve α n ( x ) ∈ U . If G is lo cally co mpact then α co mpactly co nt ractive is equiv alent with: each identit y neighbourho o d of G contains an α -inv ariant neighbo urho o d. F urther on we shall as s ume without mentioning that all g r oups are lo ca lly compact. An y conical gro up ca n b e seen as a contractible gr oup. Indee d, it suﬃces to a sso ciate to a conical gro up ( G, δ ) the contractible gro up ( G, δ ε ), for a ﬁxed ε ∈ Γ with ν ( ε ) < 1. Conv ersely , to any contractible group ( G, α ) we may asso ciate the co nical g roup ( G, δ ), with Γ =  1 2 n : n ∈ N  and for any n ∈ N and x ∈ G δ 1 2 n x = α n ( x ) . (Finally , a lo ca l conical gro up has only lo c ally the s tructure of a contractible group.) The structur e of co nt ractible gr oups is known in some detail, due to Sieb ert [27], W ang [31], Gl¨ ockner and Willis [1 6], Gl¨ ockner [15] and references therein. F or this pap er the following results ar e of interest. W e b egin with the deﬁnition of a contracting automorphism group [27], deﬁnition 5.1 . Deﬁnition 4.10 L et G b e a lo c al ly c omp act gr oup. An automorphism gr oup on G is a family T = ( τ t ) t> 0 in Aut ( G ) , such that τ t τ s = τ ts for al l t, s > 0 . The c ontr action gr oup of T is deﬁne d by C ( T ) = n x ∈ G : lim t → 0 τ t ( x ) = e o . The automorphism gr oup T is c ontr active if C ( T ) = G . 17 It is o bvious that a contractiv e a utomorphism group T induces o n G a structure of conical g roup. Conversely , any conical g roup with Γ = (0 , + ∞ ) has an asso cia ted contractiv e automorphism gro up (the gro up of dilatations ba sed at the neutral e le ment ). F urther is pro p osition 5.4 [27]. Prop ositi on 4. 11 F or a lo c al ly c omp act gr oup G the fol lo wing assertions ar e e quivalent: (i) G admits a c ontr active aut omorphism gr oup; (ii) G is a simply c onne cte d Lie gr oup whose Lie algebr a admits a p ositive gr aduatio n. 5 Other examples of dilatation structur es 5.1 Riemannian manifolds The following interesting quotation from Gr omov b o ok [17], pag es 85-86 , motiv ates so me of the ideas underlying dilatation struc tur es, espec ia lly in the very particular case of a riemannian manifold: “ 3.15. Prop osi ti on: L et ( V , g ) b e a Riema nnian manifold with g c ont inuous. F or e ach v ∈ V the sp ac es ( V , λd, v ) Lipschitz c onver ge as λ → ∞ to the tangent sp ac e ( T v V , 0) with its Euclide an metric g v . Pro of + : Start with a C 1 map ( R n , 0) → ( V , v ) whose diﬀer ential is isometric at 0. The λ -sc alings of t his pr ovide almost isometries b etwe en lar ge b al ls in R n and those in λV for λ → ∞ . Remark: In fact we c an deﬁne Riemannian manifolds as lo c al ly c omp act p ath metric sp ac es that satisfy the c onclusion of Pr op osition 3.15. “ The pro blem of domains and co doma ins left a side, a ny chart of a Riema nnian manifold induces lo cally a dilatation s tructure on the manifold. Indeed, tak e ( M , d ) to b e a n - dimensional Riemannia n manifold with d the distance o n M induced by the Riemannian structure. Consider a diﬀeomorphism φ of an op en set U ⊂ M onto V ⊂ R n and tra nsp ort the dilatations from V to U (equiv ale nt ly , tra nsp ort the distance d from U to V ). There is only one thing to chec k in order to see that we got a dilatation s tructure: the a xiom A3 , expressing the co mpatibilit y of the distance d with the dilatatio ns. But this is just a metric wa y to express the distance on the tangent space of M at x as a limit o f res c aled distances (see Gro mov Prop os itio n 3.15, [17], p. 8 5-86). Denoting by g x the metr ic tensor at x ∈ U , we have: [ d x ( u, v )] 2 = = g x  d d ε | ε =0 φ − 1 ( φ ( x ) + ε ( φ ( u ) − φ ( x ))) , d d ε | ε =0 φ − 1 ( φ ( x ) + ε ( φ ( v ) − φ ( x )))  A basica lly diﬀeren t example o f a dilatation str ucture on a riemannia n manifold will b e explained next. Let M b e a n dimensional riema nnian manifold and exp b e the geo desic exp onential. T o an y p o int x ∈ M and any vector v ∈ T x M the p oint exp x ( v ) ∈ M is lo cated on the ge o desic pass ing thru x and tangent to v ; if we parameterize this g eo desic with resp ect to length, such that the tangent at x is parallel a nd has the same direction a s v , then exp x ( v ) ∈ M has the coo rdinate equa l with the length of v with r e sp ect to the norm on T x M . W e deﬁne implicitly the dilatation based at x , of co eﬃcient ε > 0 by the relation: δ x ε exp x ( u ) = exp x ( εu ) . 18 It is not s traightforw ard to check that w e obtain a stro ng dila tation structur e, but it is true. There ar e interesting facts related to the n um bers A, B and the minimal regularity required for the riemannia n manifold. This exa mple is diﬀer ent from the ﬁrst b eca use instead of using a chart (sa me for all x ) we use a family of charts indexe d with r esp ect to the basep oint o f the dilatations . 5.2 Dilatation st ructures on the b ound ary of the dy adic tree W e shall ta ke the gr oup Γ to b e the s et of integer p ow ers o f 2 , seen as a subset o f dyadic nu m be r s. Thus for any p ∈ Z the ele ment 2 p ∈ Q 2 belo ngs to Γ. The op era tion is the m ultiplication of dyadic num b er s and the morphism ν : Γ → (0 , + ∞ ) is deﬁned by ν (2 p ) = d (0 , 2 p ) = 1 2 p ∈ (0 , + ∞ ) . The dyadic tr ee T is the inﬁnite ro o ted planar binar y tree. Any no de has tw o descen- dants. The no des are co ded by elements of X ∗ , X = { 0 , 1 } . The ro o t is co ded by the empt y word and if a no de is co ded by x ∈ X ∗ then its le ft hand s ide descendant ha s the co de x 0 and its right hand side descendant has the co de x 1. W e sha ll therefore identify the dyadic tree with X ∗ and we put on the dyadic tree the natural (ultrametric) dis ta nce on X ∗ . The bo undary (or the set of ends) of the dyadic tree is then the sa me as the co mpact ultrametric space X ω . X ω is the set of words inﬁnite at right ov er the a lphab et X = { 0 , 1 } : X ω = { f | f : N ∗ → X } = X N ∗ . A natural distance on this set is deﬁned for diﬀerent x, y ∈ X ω by the form ula d ( x, y ) = 1 2 m where m is the length of larg e st co mmon preﬁx of the words x and y . This distance is ultrametric. The metric space ( X ω , d ) is iso metric with the spa ce of dyadic integers. The metric space is then a ball of radius 1. A trivial dilatation structure is induced b y the identiﬁcation with dy adic in tegers and it has the following expressio n: δ x 2 p y = x + 2 p ( y − x ) where the o p erations are done with dyadic integers. More complex dilatation structures are given by the follo wing construction. See theorem 6.5 [7] for more details. Deﬁnition 5.1 A function W : N ∗ × X ω → I som ( X ω ) is smo oth if for any ε > 0 ther e exists µ ( ε ) > 0 su ch that for any x, x ′ ∈ X ω such t hat d ( x, x ′ ) < µ ( ε ) and for any y ∈ X ω we have 1 2 k d ( W x k ( y ) , W x ′ k ( y )) ≤ ε , for an k such that d ( x, x ′ ) < 1 / 2 k . Theorem 5.2 T o any smo oth fun ction W : N ∗ × X ω → I som ( X ω ) in t he sense of deﬁnition 5.1 is asso ciate d a dilatation stru ctur e ( X ω , d, δ ) , induc e d by functions δ x 2 , deﬁne d by δ x 2 x = x and otherwise by: for any q ∈ X ∗ , α ∈ X , x, y ∈ X ω we have δ qαx 2 q ¯ αy = q α ¯ x 1 W qαx | q | +1 ( y ) . (5.2.1) 19 5.3 Sub-rieman nian manifolds Regular sub-r iemannian manifolds provide examples of dilatation structures. In the pa pe r [8] this is explained in all details. See section 4.2 for the most basic example o f a dilatation structure on a sub-riemannia n manifold: the ca se of a Carnot g roup. More general, the dilata tion s tr uctures constructed o ver normed g roups with dilatations (theorem 4.3), w ith Γ = (0 , + ∞ ) and ν = i d , pr ovide more examples of s ub-riemannian dilatation structure s . A sub-riemannian manifold is a tr iple ( M , D , g ), wher e M is a connected manifold, D is a c ompletely non-in tegrable distr ibutio n on M , and g is a metric (Euclidean inner - pro duct) on the distribution (or horizontal bundle) D . A horizo ntal cur ve c : [ a, b ] → M is a curve which is almo st everywhere deriv able and for almo s t any t ∈ [ a, b ] w e hav e ˙ c ( t ) ∈ D c ( t ) . The class of ho r izontal cur ves is denoted by H or ( M , D ). The following theorem of Chow [11] is well known. Theorem 5.3 (Chow) L et D b e a distribution of dimension m in the manifold M . Su pp ose ther e is a p ositive int e ger numb er k (c al le d the r ank of the distribution D ) such that for any x ∈ X ther e is a top olo gic al op en b al l U ( x ) ⊂ M with x ∈ U ( x ) su ch that ther e ar e smo oth ve ctor ﬁ elds X 1 , ..., X m in U ( x ) with the pr op erty: (C) t he ve ct or ﬁ elds X 1 , ..., X m sp an D x and these ve ct or ﬁelds to gether with their iter ate d br ackets of or der at most k sp an the tangent sp ac e T y M at every p oint y ∈ U ( x ) . Then M is lo c al ly c onne cte d by horizontal curves The Carnot-Car ath´ eo dory distance (or CC distance ) asso cia ted to the sub- r iemannian manifold is the distance induced by the length l o f horiz o ntal curves: d ( x, y ) = inf { l ( c ) : c ∈ H or ( M , D ) , c ( a ) = x , c ( b ) = y } Chow co ndition (C) is used to construct an adapted frame sta rting fro m a family of vector ﬁelds which generate the distribution D . A fundamen tal result in sub-riemannian geometry is the ex is tence of normal frames. This existence result is based o n the a ccumulation of v ario us r esults by Bella ¨ ıche [2], ﬁrst to sp eak a b o ut norma l frames , providing rigo rous pro ofs for this existence in a ﬂow of res ults b etw een theorem 4.15 and ending in the ﬁrs t ha lf o f section 7 .3 (pa g e 62), Gro mov [19] in his appr oximation theorem p. 135 (conclusion of the po int (a) below), a s well in his co nvergence r esults concerning the nilpo tentization of v ector ﬁelds (related to p oint (b) b e low), V o dopy ano v and others [28] [2 9] [30] concerning the pro of of basic results in sub-r iemannian g eometry under very weak re g ularity assumptions (for a discussion of this see [8 ]). There is no place here to s ubmer ge into this, w e shall just assume that the ob ject deﬁned b elow exists. Deﬁnition 5.4 An adapte d fr ame { X 1 , ..., X n } is a normal fr ame if the fol lowing two c on- ditions ar e satisﬁe d: (a) we have the limit lim ε → 0 + 1 ε d exp n X 1 ε deg X i a i X i ! ( y ) , y ! = A ( y , a ) ∈ (0 , + ∞ ) uniformly with r esp e ct to y in c omp act sets and a = ( a 1 , ..., a n ) ∈ W , with W ⊂ R n c omp act n eighb ourho o d of 0 ∈ R n , 20 (b) for any c omp act set K ⊂ M with diameter (with re sp e ct to the distanc e d ) suﬃciently smal l, and for any i = 1 , ..., n ther e ar e fu n ctions P i ( · , · , · ) : U K × U K × K → R with U K ⊂ R n a su ﬃciently smal l c omp act neighb ourho o d of 0 ∈ R n such that for any x ∈ K and any a, b ∈ U K we have exp n X 1 a i X i ! ( x ) = exp n X 1 P i ( a, b, y ) X i ! ◦ exp n X 1 b i X i ! ( x ) and such that the fol low ing limit exists lim ε → 0 + ε − deg X i P i ( ε deg X j a j , ε deg X k b k , x ) ∈ R and it is uniform with r esp e ct to x ∈ K and a, b ∈ U K . With the help of a normal frame w e can prov e the existence of str ong dilata tion structures on regular sub-r iemannian manifolds. The following is a cons e quence of theor ems 6.3, 6.4 [8]. Theorem 5.5 L et ( M , D, g ) b e a r e gular sub-riemannian manifold, U ⊂ M an op en set which admits a normal fr ame. Deﬁn e for any x ∈ U and ε > 0 (suﬃciently smal l if ne c essary), t he dilatation δ x ε given by: δ x ε exp n X i =1 a i X i ! ( x ) ! = exp n X i =1 a i ε degX i X i ! ( x ) Then ( U, d, δ ) is a str ong dilatation stru ctur e. 6 Prop erti es of dilat ation structures 6.1 T angen t bundle A reformulation of parts of theorems 6,7 [6] is the following. Theorem 6.1 A dilatation structu r e ( X , d, δ ) has the fol lowi ng pr op erties. (a) F or al l x ∈ X , u, v ∈ X such that d ( x, u ) ≤ 1 and d ( x, v ) ≤ 1 and al l µ ∈ (0 , A ) we have: d x ( u, v ) = 1 µ d x ( δ x µ u, δ x µ v ) . We shal l say that d x has the c one pr op ert y with r esp e ct t o dilatations. (b) The metric sp ac e ( X , d ) admits a metric t angent sp ac e at x , for any p oint x ∈ X . Mor e pr e cisely we have the fol lowing limit: lim ε → 0 1 ε sup {| d ( u, v ) − d x ( u, v ) | : d ( x, u ) ≤ ε , d ( x, v ) ≤ ε } = 0 . 21 F or the next theorem (comp osite of r esults in theorems 8, 10 [6]) we need the previous ly int ro duced notion o f a nor med conical lo cal group. Theorem 6.2 L et ( X , d, δ ) b e a stro ng dilatation st ru ctur e. Then for any x ∈ X the triple ( U ( x ) , Σ x , δ x ) is a norme d lo c al c onic al gr oup, with the norm induc e d by the distanc e d x . The conica l gr oup ( U ( x ) , Σ x , δ x ) can b e rega rded as the tang ent spa ce of ( X , d, δ ) at x . F urther will b e denoted by: T x X = ( U ( x ) , Σ x , δ x ). The dilatatio n structure on this conica l group has dilata tions deﬁned by ¯ δ x,u ε y = Σ x ( u, δ x ε ∆ x ( u, y )) . (6.1.1) 6.2 T op ological considerations In this subsection we compare v ario us top olog ies and uniformities related to a dilatation structure. The axio m A3 implies tha t fo r a ny x ∈ X the function d x is contin uous, therefore op en sets with r e sp ect to d x are op en with resp ect to d . If ( X , d ) is sepa rable and d x is non deg enerate then ( U ( x ) , d x ) is als o separa ble and the top ologies of d a nd d x are the same. Ther efore ( U ( x ) , d x ) is also locally compact (and a set is compact with resp ect to d x if and only if it is compa c t with resp ect to d ). If ( X , d ) is se parable and d x is non deg enerate then the unifor mities induced b y d and d x are the same. Indeed, let { u n : n ∈ N } be a dense set in U ( x ), with x 0 = x . W e can embed ( U ( x ) , ( δ x , ε )) (see deﬁnition 7.6) isometrically in the s e parable Ba nach spa ce l ∞ , for any ε ∈ (0 , 1), by the function φ ε ( u ) =  1 ε d ( δ x ε u, δ x ε x n ) − 1 ε d ( δ x ε x, δ x ε x n )  n . A reformulation of point (a) in theorem 6.1 is that on compact sets φ ε uniformly conv erges to the iso metric embedding of ( U ( x ) , d x ) φ ( u ) = ( d x ( u, x n ) − d x ( x, x n )) n . Remark that the unifor mity induced by ( δ, ε ) is the same as the uniformity induced b y d , and that it is the same induced from the unifor mity o n l ∞ by the em bedding φ ε . W e pr ov ed that the uniformities induced by d and d x are the same. F ro m previous considerations we deduce the following characterization of tangen t spaces asso ciated to a dilatation s tructure. Corollary 6.3 L et ( X , d, δ ) b e a str ong dilatation structu r e with gr oup Γ = (0 , + ∞ ) . Then for any x ∈ X the lo c al gr oup ( U ( x ) , Σ x ) is lo c al ly a simply c onne cte d Lie gr oup whose Lie algebr a admits a p ositive gr aduation (a homo gene ous gr oup). Pro of. Use the facts: ( U ( x ) , Σ x ) is a lo cally c ompact group (from previous topo logical consideratio ns) which admits δ x as a contractive automorphism g roup (from theorem 6.2). Apply then Sieb er t pro p o sition 4.11 ( which is [27] prop osition 5 .4).  22 6.3 Diﬀeren t iabilit y with resp ect to dilatation structures W e brieﬂy explain the notion of diﬀerentiabilit y asso ciated to dilata tio n structures (sec tio n 7.2 [6]). Firs t we need the na tur al deﬁnition b elow. Deﬁnition 6.4 L et ( N , δ ) and ( M , ¯ δ ) b e two c onic al gr oups. A function f : N → M is a c onic al gr oup morphism if f is a gr oup morphism and for any ε > 0 and u ∈ N we have f ( δ ε u ) = ¯ δ ε f ( u ) . The deﬁnition of the der iv ative with resp ect to dilatations structures follows. Deﬁnition 6.5 L et ( X , δ, d ) and ( Y , δ , d ) b e two str ong dilatation structur es and f : X → Y b e a c ontinuous function. The function f is diﬀer ent iable in x if ther e exist s a c onic al gr oup morphism Q x : T x X → T f ( x ) Y , deﬁne d on a neighb ourho o d of x with values in a neighb ourho o d of f ( x ) such that lim ε → 0 sup  1 ε d  f ( δ x ε u ) , δ f ( x ) ε Q x ( u )  : d ( x, u ) ≤ ε  = 0 , (6.3.2) The morphism Q x is c al le d the derivative of f at x and wil l b e sometimes denote d by D f ( x ) . The function f is u niformly diﬀer ent iable if it is diﬀer entiable everywher e and the limit in (6.3.2) is u niform in x in c omp act s ets. 7 Inﬁnitesimal aﬃne geometry of dilatation structures 7.1 Aﬃne t ransformations Deﬁnition 7.1 L et ( X , d, δ ) b e a dilatation str u ctur e. A tr ansformation A : X → X is aﬃne if it is Lipschitz and it c ommutes with dilatations in the fol lowing sense: for any x ∈ X , u ∈ U ( x ) and ε ∈ Γ , ν ( ε ) < 1 , if A ( u ) ∈ U ( A ( x )) then Aδ x ε = δ A ( x ) ε A ( u ) . The lo c al gr oup of aﬃne tr ansformations, denote d by Af f ( X , d, δ ) is forme d by al l invertible and bi-lipschitz aﬃne tr ansformations of X . Af f ( X, d, δ ) is indeed a lo c a l gr oup. In order to see this we star t from the remar k that if A is Lipschitz then there exists C > 0 such that for all x ∈ X and u ∈ B ( x, C ) we have A ( u ) ∈ U ( A ( x )). The in v erse of A ∈ Af f ( X , d, δ ) is then aﬃne. Sa me considerations apply for the co mp o sition of tw o aﬃne, bi-lipschitz and inv ertible transformatio ns . In the particula r ca se o f X ﬁnite dimensional rea l, nor med vector space, d the distanc e given b y the norm, Γ = (0 , + ∞ ) and dilatations δ x ε u = x + ε ( u − x ), an aﬃne transforma tion in the sense o f deﬁnition 7.1 is a n aﬃne tr ansformatio n of the vector space X . Prop ositi on 7. 2 L et ( X, d, δ ) b e a dilatation struct ur e and A : X → X an aﬃne tr ansfor- mation. Then: (a) for al l x ∈ X , u, v ∈ U ( x ) suﬃciently close to x , we have: A Σ x ε ( u, v ) = Σ A ( x ) ε ( A ( u ) , A ( v )) . 23 (b) for al l x ∈ X , u ∈ U ( x ) suﬃciently close to x , we have: A inv x ( u ) = inv A ( x ) A ( u ) . Prop ositi on 7. 3 L et ( X , d, δ ) b e a str ong dilatation structur e and A : X → X an aﬃne tr ansformation. Then A is uniformly diﬀer entiable and the derivative e quals A . The pro o fs are straig htf orward, just use the co mm utation with dilatations. 7.2 Inﬁnitesimal linearity of dilatation struct ures W e b egin by an explana tion o f the term ” suﬃciently clos e d“ , which will be used rep eatedly in the following. W e w ork in a dila ta tion structure ( X , d, δ ). Let K ⊂ X b e a co mpa ct, non empty set. Then there is a constant C ( K ) > 0 , depending on the set K such that for any ε , µ ∈ Γ with ν ( ε ) , ν ( µ ) ∈ (0 , 1 ] and a ny x, y , z ∈ K with d ( x, y ) , d ( x, z ) , d ( y , z ) ≤ C ( K ) we hav e δ y µ z ∈ V ε ( x ) , δ x ε z ∈ V µ ( δ x ε y ) . Indeed, this is coming fro m the uniform (with re sp ect to K ) estimates: d ( δ x ε y , δ x ε z ) ≤ ε d x ( y , z ) + ε O ( ε ) , d ( x, δ y µ z ) ≤ d ( x, y ) + d ( y , δ y µ z ) ≤ d ( x, y ) + µd y ( y , z ) + µ O ( µ ) . Deﬁnition 7.4 A pr op erty P ( x 1 , x 2 , x 3 , ... ) holds for x 1 , x 2 , x 3 , ... suﬃciently close d if for any c omp act, non empty set K ⊂ X , ther e is a p ositive c onstant C ( K ) > 0 su ch t hat P ( x 1 , x 2 , x 3 , ... ) is true for any x 1 , x 2 , x 3 , ... ∈ K with d ( x i , x j ) ≤ C ( K ) . F or example, we may say that the expres sions δ x ε δ y µ z , δ δ x ε y µ δ x ε z are w ell deﬁned for any x, y , z ∈ X suﬃciently closed and for an y ε, µ ∈ Γ with ν ( ε ) , ν ( µ ) ∈ (0 , 1]. Deﬁnition 7.5 A dila tation structur e ( X , d, δ ) is line ar if for any ε , µ ∈ Γ su ch that ν ( ε ) , ν ( µ ) ∈ (0 , 1 ] , and for any x, y , z ∈ X suﬃciently close d we have δ x ε δ y µ z = δ δ x ε y µ δ x ε z . This deﬁnition means simply that a linear dila tation s tr ucture is a dilata tion structur e with the pr op erty that dilatations are aﬃne in the s ense of deﬁnition 7.1. Let us lo ok to a dilatation structure in ﬁner details. W e do this by deﬁning induced dilatation structure s from a g iven o ne. Deﬁnition 7.6 L et ( X , δ, d ) b e a dilatation stru ctur e and x ∈ X a p oint. In a neighb ourho o d U ( x ) of x , for any µ ∈ (0 , 1) we deﬁne the distanc es: ( δ x , µ )( u, v ) = 1 µ d ( δ x µ u, δ x µ v ) . 24 The next theorem shows that on a dila tation structure we almost hav e translatio ns (the op erator s Σ x ε ( u, · )), which are almost isometries (that is, not with resp ect to the distance d , but with r esp ect to distances of type ( δ x , µ )). It is almost as if we were working with a nor med conical gr oup, only that we have to use fa milies of distances and to make s ma ll shifts in the tangent space, as it is done a t the end of the pro of of theorem 7 .7. Theorem 7.7 L et ( X , δ, d ) b e a (str ong) dilatation structur e. F or any u ∈ U ( x ) and v close to u let u s deﬁne ˆ δ x,u µ,ε v = Σ x µ ( u, δ δ x µ u ε ∆ x µ ( u, v )) = δ x µ − 1 δ δ x µ u ε δ x µ v . Then ( U ( x ) , ˆ δ x µ , ( δ x , µ )) is a (str ong) dilatation stru ctur e. The tr ans formation Σ x µ ( u, · ) is an isometry fr om ( δ δ x µ u , µ ) to ( δ x , µ ) . Mor e over, we have Σ x µ ( u, δ x µ u ) = u . Pro of. W e hav e to chec k the axioms. The ﬁrst par t o f axio m A0 is a n easy cons e quence of theore m 6.1 for ( X , δ, d ). The seco nd part o f A0 , A1 a nd A2 are true ba s ed on simple computations. The ﬁrst interesting fact is related to axio m A3. Let us compute, for v , w ∈ U ( x ), 1 ε ( δ x , µ )( ˆ δ x,u µ,ε v , ˆ δ x,u µε w ) = 1 εµ d ( δ x µ ˆ δ x,u µε v , δ x µ ˆ δ x,u µε w ) = = 1 εµ d ( δ δ x µ u ε δ x µ v , δ δ x µ u ε δ x µ w ) = 1 εµ d ( δ δ x µ u εµ ∆ x µ ( u, v ) , δ δ x µ u εµ ∆ x µ ( u, w )) = = ( δ δ x µ u , εµ )(∆ x µ ( u, v ) , ∆ x µ ( u, w )) . The axiom A3 is then a consequenc e of axiom A3 for ( X , d, δ ) and we ha ve lim ε → 0 1 ε ( δ x , µ )( ˆ δ x,u µε v , ˆ δ x,u µε w ) = d δ x µ u (∆ x µ ( u, v ) , ∆ x µ ( u, w )) . The axiom A4 is als o a straightforward conseque nc e of A4 fo r ( X , d, δ ). The second par t o f the theorem is a simple computation.  The induced dilatation s tr uctures ( U ( x ) , ˆ δ x µ , ( δ x , µ )) should conv erge in some sense to the dilatation structure on the tange nt space a t x , a s ν ( µ ) conv erges to zer o. Remar k that we hav e one easy co nvergence in strong dilatation s tructures: lim µ → 0 ˆ δ x,u µ,ε v = ¯ δ x,u ε v where ¯ δ x are the dilatatio ns in the tangent space at x , cf. (6.1 .1). Indeed, this c o mes from: ˆ δ x,u µ,ε v = Σ x µ ( u, δ δ x µ u ε ∆ x µ ( u, v )) so, when ν ( µ ) c onv erges we get the mentioned limit. The following prop ositio n g ives a more pr ecise estimate: the order o f approximation of the dilatations δ b y dilatations ˆ δ x ε , in neighbourho o ds of δ x ε y of order ε , as ν ( ε ) go es to zero. Prop ositi on 7. 8 L et ( X , δ, d ) b e a dilatation stru ct ur e. With the notations of the or em 7.7 we intr o duc e ˆ δ x,u ε v = ˆ δ x,u ε,ε v = δ x ε − 1 δ δ x ε u ε δ x ε v . Then we have for any x, y , v suﬃciently close d: lim ε → 0 1 ε ( δ x , ε )  δ δ x ε y ε v , ˆ δ x,δ x ε y ε v  = 0 . (7.2.1) 25 Pro of. W e star t by a computation: 1 ε ( δ x , ε )  δ δ x ε y ε v , ˆ δ x,δ x ε y ε v  = 1 ε 2 d  δ x ε δ δ x ε y ε v , δ x ε ˆ δ x,δ x ε y ε v  = = 1 ε 2 d  δ x ε 2 Σ x ε ( y , v ) , δ x ε 2 δ x ε − 2 δ δ x ε 2 y ε 2 ∆ x ε ( δ x ε y , v )  = = 1 ε 2 d ( δ x ε 2 Σ x ε ( y , v ) , δ x ε 2 Σ x ε 2 ( y , ∆ x ε ( δ x y , v ))) . This last expre s sion conv erges as ν ( ε ) go es to 0 to d x (Σ x ( y , v ) , Σ x ( y , ∆ x ( x, v ))) = d x ( v , ∆ x ( x, v )) = 0  The result from this propo sition indicates that strong dila tation structures are inﬁnites- imally linear. In o rder to make a pr ecise statement w e need a measure for nonlinea rity of a dilatation structure, given in the next deﬁnition. Then we hav e to rep eat the computations from the pro o f of pro p o sition 7.8 in a slightly diﬀerent s etting, related to this measur e of nonlinearity . Deﬁnition 7.9 The fol lowing expr ession: Lin ( x, y , z ; ε , µ ) = d  δ x ε δ y µ z , δ δ x ε y µ δ x ε z  (7.2.2) is a me asur e of lack of line arity, for a gener al dilatation st ructur e. The next theor em shows that indeed, inﬁnitesimally any s tr ong dilatation str uctur e is linear. Theorem 7.10 L et ( X, d, δ ) b e a str ong dilatation struct ur e. Then for any x, y , z ∈ X suﬃciently close we have lim ε → 0 1 ε 2 Lin ( x, δ x ε y , z ; ε, ε ) = 0 . (7.2.3) Pro of. F rom the hypothesis of the theorem we hav e: 1 ε 2 Lin ( x, δ x ε y , δ x ε z ; ε, ε ) = 1 ε 2 d  δ x ε δ δ x ε y ε z , δ δ x ε 2 y ε δ x ε z  = = 1 ε 2 d  δ x ε 2 Σ x ε ( y , z ) , δ x ε 2 δ x ε − 2 δ δ x ε 2 y ε δ x ε z  = = 1 ε 2 d ( δ x ε 2 Σ x ε ( y , z ) , δ x ε 2 Σ x ε 2 ( y , ∆ x ε ( δ x ε y , z ))) = = O ( ε 2 ) + d x (Σ x ε ( y , z ) , Σ x ε 2 ( y , ∆ x ε ( δ x ε y , z ))) . The dilatation structur e satisﬁes A4, therefor e as ε g o es to 0 we o btain: lim ε → 0 1 ε 2 Lin ( x, δ x ε y , δ x ε z ; ε, ε ) = d x (Σ x ( y , z ) , Σ x ( y , ∆ x ( x, z ))) = = d x (Σ x ( y , z ) , Σ x ( y , z )) = 0 .  26 7.3 Linear strong dilatation structures Remark that for gener al dilatation structures the ”translatio ns” ∆ x ε ( u, · ) are not aﬃne. Nevertheless, they co mm ute with dilatation in a known wa y: for any u, v suﬃciently close to x and µ ∈ Γ , ν ( µ ) < 1 , we hav e : ∆ x ε  δ x µ u, δ x µ v  = δ δ x ǫµ u µ ∆ x εµ ( u, v ) . This is imp ortant, b eca us e the tra nsformations Σ x ε ( u, · ) really b ehave as translatio ns. The reason for which such tr ansformatio ns ar e not aﬃne is that dilata tions are genera lly not aﬃne. Linear dilatatio n structures a re very particular dilatatio n structures. The next prop osi- tion gives a family of e xamples of linear dilatation structures. Prop ositi on 7. 11 The dilatation structu r e asso ciate d to a n orme d c onic al gr oup is line ar. Pro of. Indeed, for the dilatation structure asso ciated to a normed conical group we ha ve, with the no tations from deﬁnition 7.5: δ δ x ε y µ δ x ε z =  xδ ε ( x − 1 y )  δ µ  δ ε ( y − 1 x ) x − 1 x δ ε ( x − 1 z )  = =  xδ ε ( x − 1 y )  δ µ  δ ε ( y − 1 x ) δ ε ( x − 1 z )  =  xδ ε ( x − 1 y )  δ µ  δ ε ( y − 1 z )  = = x  δ ε ( x − 1 y ) δ ε δ µ ( y − 1 z )  = x δ ε  x − 1 y δ µ ( y − 1 z )  = δ x ε δ y µ z . Therefore the dila tation structure is linear .  The aﬃnity of tra nslations Σ x ε is related to the linearity of the dilatation s tr ucture, as describ ed in the theorem b elow, p oint (a). As a consequence, we prove at p o int (b) that a linear and str ong dilatation structur e comes fro m a co nical gro up. Theorem 7.12 L et ( X , d, δ ) b e a dilatation st ru ctur e. (a) If the dilatation structure is line ar then al l tr ansformations ∆ x ε ( u, · ) ar e aﬃne for any u ∈ X . (b) If the dilatation structu r e is st r ong (satisﬁes A4) then it is line ar if and only if the dilatations c ome fr om t he dilatation structur e of a c onic al gr ou p, pr e cisely for any x ∈ X ther e is an op en neighb ourho o d D ⊂ X of x su ch that ( D , d x , δ ) is the s ame dilatation st ructur e as the dilatation structu r e of t he tangent sp ac e of ( X , d, δ ) at x . Pro of. (a) If dilatations are a ﬃne, then let ε, µ ∈ Γ, ν ( ε ) , ν ( µ ) ≤ 1, and x, y , u , v ∈ X such that the following computations make sense. W e hav e: ∆ x ε ( u, δ y µ v ) = δ δ x ε u ε − 1 δ x ε δ y µ v . Let A ε = δ δ x ε u ε − 1 . W e compute: δ ∆ x ε ( u,y ) µ ∆ x ε ( u, v ) = δ A ε δ x ε y µ A ε δ x ε v . W e use twice the aﬃnity of dilatations: δ ∆ x ε ( u,y ) µ ∆ x ε ( u, v ) = A ε δ δ x ε y µ δ x ε v = δ δ x ε u ε − 1 δ x ε δ y µ v . 27 W e prov ed that: ∆ x ε ( u, δ y µ v ) = δ ∆ x ε ( u,y ) µ ∆ x ε ( u, v ) , which is the c o nclusion of the part (a). (b) Supp ose that the dilata tion structure is strong. If dilatations are aﬃne, then by po int (a) the tra nsformations ∆ x ε ( u, · ) are aﬃne as well fo r any u ∈ X . Then, with notatio ns made b efore, for y = u we get ∆ x ε ( u, δ u µ v ) = δ δ x ε u µ ∆ x ε ( u, v ) , which implies δ u µ v = Σ x ε ( u, δ x µ ∆ x ε ( u, v )) . W e pass to the limit with ε → 0 and we obtain: δ u µ v = Σ x ( u, δ x µ ∆ x ( u, v )) . W e recog nize at the right hand side the dilatations a sso ciated to the conica l group T x X . By prop ositio n 7 .11 the oppo site implication is straightforward, b ec a use the dilata tion structure of any conical g roup is linear .  8 Noncomm utativ e aﬃne geometry W e prop ose her e to call ”noncommutativ e aﬃne geometry “ the g e ne r alization of aﬃne ge- ometry descr ib e d in theorem 2.5, but without the restrictio n Γ = (0 , + ∞ ). F or shor t, noncommutativ e aﬃne g eometry in the sense explained further is the study of the prop er- ties of linear s trong dilatation struc tur es. Equa lly , by theorem 7.1 2, it is the study of normed conical groups. As a mo tiv ation for this name, in the prop osition below w e give a relatio n, true for linear dilatation s tructures, with a n int eresting interpretation. W e shall explain what this relation mea ns in the most trivial case: the dilatation structure asso ciated to a real normed aﬃne spa ce. In this ca se, for a ny points x, u, v , let us denote w = Σ x ε ( u, v ). Then w equals (approximativ ely , due to the parameter ε ) the sum u + x v . Denote also w ′ = ∆ u ε ( x, v ); then w ′ is (approximatively) equal to the diﬀerence be tw een v and x based at u . In o ur space (a classical aﬃne s pace o v er a vector s pa ce) we ha ve w = w ′ . The next prop osition shows that the same is true for any linea r dilatation s tructure. Prop ositi on 8. 1 F or a line ar dilatation stru ctur e ( X , δ, d ) , for any x, u, v ∈ X suﬃciently close d and for any ε ∈ Γ , ν ( ε ) ≤ 1 , we have: Σ x ε ( u, v ) = ∆ u ε ( x, v ) . Pro of. W e have the following string o f equalities, by using twice the linear ity of the di- latation structure : Σ x ε ( u, v ) = δ x ε − 1 δ δ x ε u ε v = δ u ε δ x ε − 1 v = = δ δ u ε x ε − 1 δ u ε v = ∆ u ε ( x, v ) . The pro of is done.  28 8.1 In v erse semigroups and Menelaos theorem Here we prove that for s trong dila tation structures line a rity is equiv alent to a genera lization of the statement fro m cor ollary 2.8. The result is new fo r Ca rnot gro ups a nd the pro o f s e ems to b e new even for vector s paces. Deﬁnition 8.2 A semigr oup S is an inverse semigr oup if for any x ∈ S ther e is an unique element x − 1 ∈ S such that x x − 1 x = x and x − 1 x x − 1 = x − 1 . An imp or tant example o f an inv erse semigro up is I ( X ), the class of all bijective maps φ : dom φ → i m φ , with dom φ, im φ ⊂ X . The s e migroup op er ation is the comp osition of functions in the larges t domain where this makes sense. By the V agner- Preston repr esentation theorem [23] every inv erse semigr oup is iso morphic to a subsemigr oup of I ( X ), for some set X . Deﬁnition 8.3 A dilatation st ructur e ( X , d, δ ) has the Menelaos pr op ert y if for any two suﬃciently close d x, y ∈ X and for any ε, µ ∈ Γ with ν ( ε ) , ν ( µ ) ∈ (0 , 1) we have δ x ε δ y µ = δ w εµ , wher e w ∈ X is the ﬁ xe d p oint of the c ontra ction δ x ε δ y µ (thus dep ending on x, y and ε, µ ). Theorem 8.4 A line ar dilatation s t ructur e has the Menelaos pr op erty. Pro of. Let x, y ∈ X be suﬃciently closed and ε, µ ∈ Γ with ν ( ε ) , ν ( µ ) ∈ (0 , 1 ). W e shall deﬁne tw o sequences x n , y n ∈ X , n ∈ N . W e b e gin with x 0 = x , y 0 = y . Supp o se further that x n , y n were deﬁned and that they are suﬃciently closed. Then we use twice the linearity of the dilatation structure: δ x n ε δ y n µ = δ δ x n ε y n µ δ x n ε = δ δ δ x n ε y n µ x n ε δ δ x n ε y n µ . W e shall deﬁne then by induction x n +1 = δ δ x n ε y n µ x n , y n +1 = δ x n ε y n . (8.1.1) Provided that we prov e by induction that x n , y n are suﬃciently closed, we arr ive at the conclusion that for any n ∈ N δ x n ε δ y n µ = δ x ε δ y µ . (8.1.2 ) The p oints x 0 , y 0 are suﬃciently closed by hypothesis. Supp ose now that x n , y n are suﬃ- ciently closed. Due to the linea rity of the dila tation structure, w e ca n wr ite the ﬁr st pa rt of (8.1.1) as: x n +1 = δ x n ε δ y n µ x n . Then we can estimate the dis ta nce b etw een x n +1 , y n +1 like this: d ( x n +1 , y n +1 ) = d ( δ x n ε δ y n µ x n , δ x n ε y n ) = ν ( ε ) d ( δ y n µ x n , y n ) = ν ( εµ ) d ( x n , y n ) . F ro m ν ( εµ ) < 1 it follows tha t x n +1 , y n +1 are suﬃciently closed. By induction we deduce that for a ll n ∈ N the p oints x n +1 , y n +1 are suﬃciently closed. W e als o ﬁnd out tha t lim n →∞ d ( x n , y n ) = 0 . (8.1.3) 29 F ro m relation (8.1 .2) we deduce that the ﬁrst part o f (8.1.1) can b e written as : x n +1 = δ x n ε δ y n µ x n = δ x ε δ y µ x n . The transforma tion δ x ε δ y µ is a contraction of coeﬃcie nt ν ( εµ ) < 1, therefore we easily get: lim n →∞ x n = w , (8.1.4) where w is the unique ﬁxed p oint of the contraction δ x ε δ y µ . W e put together (8.1.3) a nd (8 .1.4) and we get the limit: lim n →∞ y n = w , (8.1.5) Using relations (8.1 .4), (8.1.5), we may pass to the limit with n → ∞ in re la tion (8.1.2): δ x ε δ y µ = lim n →∞ δ x n ε δ y n µ = δ w ε δ w µ = δ w εµ . The pro of is done.  Corollary 8.5 L et ( X, d, δ ) b e a str ong line ar dilatation structur e, with gr oup Γ and the morphism ν inje ct ive. Then any element of the inverse subsemigr oup of I ( X ) gener ate d by dilatations is lo c al ly a dilatation δ x ε or a left t r anslation Σ x ( y , · ) . Pro of. Let ( X , d, δ ) b e a str o ng linear dilatation structure. F rom the linear ity and theorem 8.4 we deduce that we hav e to care only about the results of comp ositions o f tw o dilatations which a r e isometries . The dilatation structure is strong, therefore by theo r em 7.1 2 the dilatation s tr ucture is lo cally coming fro m a co nical gro up. Let us compute a comp osition of dilatations δ x ε δ y µ , with ν ( εµ ) = 1. Because the mor phism ν is injective, it follows that µ = ε − 1 . In a conical gro up we can make the following computation (here δ ε = δ e ε with e the neutral element of the conical gr o up): δ x ε δ y ε − 1 z = xδ ε  x − 1 y δ ε − 1  y − 1 z  = xδ ε  x − 1 y  y − 1 z . Therefore the co mpo sition of dilatatio ns δ x ε δ y µ , with εµ = 1, is a left transla tion. Another ea s y computation shows that comp os itio n of left translatio ns with dila tations are dilatations. The pro o f end by remarking that a ll the statements are lo ca l.  A coun terexample. The Corolla r y 8.5 is not true without the injectivit y assumption on ν . Indeed, co nsider the Car not gr oup N = C × R with the elements denoted by X ∈ N , X = ( x, x ′ ), with x ∈ C , x ′ ∈ R , and o p eration X Y = ( x, x ′ )( y , y ′ ) = ( x + y , x ′ + y ′ + 1 2 I m x ¯ y ) W e take Γ = C ∗ and morphism ν : Γ → (0 , + ∞ ), ν ( ε ) = | ε | . Dilatations are deﬁned as : for any ε ∈ C ∗ and X = ( x, x ′ ) ∈ N : δ ε X = ( εx, | ε | 2 x ′ ) These dilatations induce the ﬁeld of dilata tions δ X ε Y = X δ ε ( X − 1 Y ). The morphis m ν is not injective. Let now ε, µ ∈ C ∗ with εµ = − 1 and ε ∈ (0 , 1 ). An elementary (but a bit long) computation shows that for X = (0 , 0) and Y = ( y , y ′ ) with y 6 = 0, y ′ 6 = 0, the comp osition of dilatations δ X ε δ Y µ is not a left transla tion in the g roup N , nor a dila tation.  F urther we sha ll supp ose that the morphism ν is alwa ys injective, if not explicitly stated otherwise. Ther e fore we s ha ll consider Γ ⊂ (0 , + ∞ ) as a subgroup. 30 8.2 On t he barycen tric condition The bar ycentric condition is (Af3): for any ε ∈ (0 , 1) δ x ε y = δ y 1 − ε x . In par ticular, the condition (Af3) tells that the transfor mation y 7→ δ y ε x is also a dilatation. Is this true for linear dilatation structures? Theo rem 2.5 indicates that (Af3) is true if and only if this is a dilatation structure of a normed rea l aﬃne space. Prop ositi on 8. 6 L et X b e a n orme d c onic al gr oup with neut r al element e , dilatations δ and distanc e d induc e d by the homo gene ous norm k · k , and ε ∈ (0 , 1) ∩ Γ . Then the funct ion h ε : X → X , h ε ( x ) = xδ ε ( x − 1 ) = δ x ε e is invertible and the inverse g ε has the expr ession g ε ( y ) = ∞ Y k =0 δ ε k ( y ) = lim N →∞ N Y k =0 δ ε k ( y ) Remark 8.7 As the choic e of the neut r al element is not imp ortant , the pr evious pr op osition says that for any ε ∈ (0 , 1 ) and any ﬁxe d y ∈ X t he fun ct ion x 7→ δ x ε y is invertible. Pro of. Let ε ∈ (0 , 1) b e ﬁxed. F or any natural num b er N we deﬁne g N : X → X by g N ( y ) = N Y k =0 δ ε k ( y ) F or ﬁxed y ∈ X ( g N ( y )) N is a Cauch y sequence. Indeed, for any N ∈ N we hav e: d ( g N ( y ) , g N +1 ( y )) = k δ ε N +1 ( y ) k th us for any N , M ∈ N , M ≥ 1 we ha ve d ( g N ( y ) , g N + M ( y )) ≤ M X k = N +1 ε k ! k y k ≤ ε N +1 1 − ε k y k Let then g ε ( y ) = lim N →∞ g N ( y ). W e prov e that g ε is the inv er se of h ε . W e hav e, for a ny natural num b er N and y ∈ X y δ ε g N ( y ) = g N +1 ( y ) By passing to the limit with N w e get that h ε ◦ g ε ( y ) = y for any y ∈ X . Let us now compute g N ◦ h ε ( x ) = N Y k =0 δ ε k ( xδ ε ( x − 1 )) = N Y k =0 δ ε k ( x ) δ ε k +1 ( x − 1 ) = = x δ ε N +1 ( x − 1 ) therefore as N go es to inﬁnity we get g ε ◦ h ε ( x ) = x .  F or any ε ∈ (0 , 1 ) the functions h ε , g ε are homog eneous, that is h ε ( δ µ x ) = δ µ h ε ( x ) , g ε ( δ µ y ) = δ µ g ε ( y ) for any µ > 0 and x, y ∈ X . In the presence of the baryce ntric condition we get the following: 31 Corollary 8.8 L et ( X , d, δ ) b e a str ong dilatatio n stru ctur e with gr oup Γ ⊂ (0 , + ∞ ) , which satisﬁes the b aryc entric c ondition (A f3). Then for any u, v ∈ X and ε ∈ (0 , 1) ∩ Γ the p oints inv u ( v ) , u and δ u ε v ar e c ol line ar in the sense: d ( inv u ( v ) , u ) + d ( u, δ u ε v ) = d ( inv u ( v ) , δ u ε v ) Pro of. There is no res triction to work with the group o p e ration with neutra l element e and denote δ ε := δ e ε . With the nota tion from the pro of of the pr op osition 8.6, w e use the expression of the function g ε , we apply the homogeneo us nor m k · k and we obtain: k g ε ( y ) k ≤ ∞ X k =0 ε k ! k x k = 1 1 − ε k y k with equality if and only if e , y a nd y δ ε y are c o llinear in the sense d ( e, y ) + d ( y , y δ ε y ) = d ( e, y δ ε y ). The baryce nt ric condition can be wr itten as : h ε ( x ) = δ 1 − ε ( x ). W e have therefore: k x k = k g ε ◦ h ε ( x ) k ≤ 1 1 − ε k h ε ( x ) k = 1 − ε 1 − ε k x k = k x k therefore e , x and xδ ε x are on a geo desic. This is tr ue also for the choice: e = in v u ( v ), x = u , which g ives the conclusion.  W e can actually say more in the ca se Γ = (0 , + ∞ ). Prop ositi on 8. 9 L et ( X , d, δ ) b e a s t r ong dilatation stru ctur e with gr oup Γ = (0 , + ∞ ) , which satisﬁes the b aryc entric c ondition (Af3). Then for any x ∈ X t he gr oup op er ation Σ x is ab elia n and mor e over the gr aduation of X , as a homo gene ous gr oup with r esp e ct to the op er ation Σ x has only one level. Pro of. Let us deno te the neutral element by e instead of x and denote δ ε := δ e ε . According to cor ollary 6.3 X is a Lie homogeneous group. The barycentric condition implies: for a ny x ∈ X a nd ε ∈ (0 , 1) we hav e δ 1 − ε y = y δ ε y − 1 , which implies: δ 1 − ε ( y ) δ ε ( y ) = y for a ny y and for an y ε ∈ (0 , 1). This fact implies that { δ µ y : µ ∈ (0 , + ∞ ) } is a one parameter semigro up. Moreover, let f y : R → X , deﬁned b y: if ε > 0 then f y ( ε ) = δ ε y , else f y ( ε ) = δ ε y − 1 . Then f y is a gro up mor phis m fr om R to X , with f y (1) = y . Therefor e f y ( ε ) = exp( εy ) = εy . Acco rding to deﬁnition 2.3 the gro up X is identiﬁed with its Lie algebra and any ele ment y has a decomp ositio n y = y 1 + y 2 + ... + y m and δ ε y = m X j =1 ε j y j . W e prov ed that m = 1, o therwise said that the gr a duation of the group ha s only one level, that is the group is ab elian.  8.3 The r at io of three collinear p oin ts In this section we pr ov e that the nonco mmutative aﬃne geometry is a geometry in the s e nse of the Er langen progra m, b ecause it can b e describ ed as the geometry of collinear triples (see deﬁnition 8.10). Collinear triples gener alize the ba sic r atio inv ariant of clas s ical aﬃne geometry . 32 Indeed, theorem 8.4 provides us with a mean to in tro duce a v ersion of the r atio of three collinear p oints in a str ong linear dilatatio n structure. W e deﬁne here c ol line ar triples , the r atio fun ction and the r atio n orm . Deﬁnition 8.10 L et ( X, d, δ ) b e a st r ong line ar dilatation structure . Denote by x α = ( x, α ) , for any x ∈ X and α ∈ (0 , + ∞ ) . An or der e d set ( x α , y β , z γ ) ∈ ( X × (0 , + ∞ )) 3 is a c ol line ar triple if: (a) αβ γ = 1 and al l thr e e numb ers ar e diﬀer ent fr om 1 , (b) we have δ x α δ y β δ z γ = id . The r atio norm r ( x α , y β , z γ ) of the c ol line ar triple ( x α , y β , z γ ) is given by the expr ession: r ( x α , y β , z γ ) = α 1 − αβ L et ( x α , y β , z γ ) b e a c ol line ar triple. Then we have: δ x α δ y β = δ z αβ with α, β , αβ not e qual to 1 . By the or em 8.4 the p oint z is u niquely determine d by ( x α , y β ) , ther efor e we c an expr ess it as a function z = w ( x, y , α, β ) . The function w is c al le d the r atio function. In the next pr op osition we o btain a formula fo r w ( x, y , α, β ). Alternatively this can b e seen as ano ther pro of of theorem 8.4. Prop ositi on 8. 11 In the hyp othesis of pr op osition 8.6, for any ε, µ ∈ (0 , 1 ) and x, y ∈ X we have: w ( x, y , ε, µ ) = g εµ ( h ε ( x ) h µ ( δ ε y )) Pro of. Any z ∈ X with the pro p er ty that for a ny u ∈ X we have δ x ε δ y µ ( u ) = δ z εµ ( u ) satisﬁes the equa tion: x δ ε  x − 1 y δ µ ( y − 1 )  = z δ εµ ( z − 1 ) (8.3.6) This equation ca n be put as: h ε ( x ) δ ε ( h µ ( y )) = h εµ ( z ) F ro m prop o sition 8.6 we obtain that indeed exis ts and it is unique z ∈ X solution of this equation. W e use further homoge ne ity of h µ and we get: z = w ( x, y , ε, µ ) = g εµ ( h ε ( x ) h µ ( δ ε y ))  Remark that if ( x α , y β , z γ ) is a collinear triple then any circular pe rmutation of the triple is also a co lline a r triple. W e can not deduce from here a c ollinearity no tio n for the triple of p oints { x, y , z } . Indeed, as the following example s hows, even if ( x α , y β , z γ ) is a collinear triple, it may happ en that here are no num ber s α ′ , β ′ , γ ′ such that ( y β ′ , x α ′ , z γ ′ ) is a c o llinear triple. 33 Colline ar triples i n the H e isenberg g roup. The Heisenberg gro up H ( n ) = R 2 n +1 is a 2- step Car not g r oup. F or the p oints of X ∈ H ( n ) we use the nota tion X = ( x, ¯ x ), with x ∈ R 2 n and ¯ x ∈ R . The group op er ation is : X Y = ( x, ¯ x )( y , ¯ y ) = ( x + y , ¯ x + ¯ y + 1 2 ω ( x, y )) where ω is the standard symplectic form on R 2 n . W e s hall iden tify the Lie a lg ebra with the Lie group. The brack et is [( x, ¯ x ) , ( y , ¯ y )] = (0 , ω ( x, y )) The Heisenberg a lgebra is gener ated by V = R 2 n × { 0 } and we ha v e the rela tions V + [ V , V ] = H ( n ), { 0 } × R = [ V , V ] = Z ( H ( n )). The dilatatio ns on H ( n ) are δ ε ( x, ¯ x ) = ( εx, ε 2 ¯ x ) F or X = ( x, ¯ x ) , Y = ( y , ¯ y ) ∈ H ( n ) a nd ε, µ ∈ (0 , + ∞ ), εµ 6 = 1, we compute Z = ( z , ¯ z ) = w ( ˜ x , ˜ y , ε , µ ) with the help of equation (8.3 .6). This equatio n writes: ((1 − ε ) x, (1 − ε 2 ) ¯ x ) ( ε (1 − µ ) y , ε 2 (1 − µ 2 ) ¯ y ) = ((1 − εµ ) z , (1 − ε 2 µ 2 ) ¯ z ) After using the express ion of the gr o up op eration we obtain: Z =  1 − ε 1 − εµ x + ε (1 − µ ) 1 − εµ y , 1 − ε 2 1 − ε 2 µ 2 ¯ x + ε 2 (1 − µ 2 ) 1 − ε 2 µ 2 ¯ y + ε (1 − ε )(1 − µ ) 2(1 − ε 2 µ 2 ) ω ( x, y )  Suppo se now that ( X α , Y β , Z γ ) and ( Y β ′ , X α ′ , Z γ ′ ) are collinear triples suc h that X = ( x, 0), Y = ( y , 0 ) and ω ( x, y ) 6 = 0. F r o m the computation of the ratio function, we get that there exist num bers k , k ′ 6 = 0 such that: z = k x + (1 − k ) y = (1 − k ′ ) x + k ′ y , ¯ z = k (1 − k ) 2 ω ( x, y ) = k ′ (1 − k ′ ) 2 ω ( y , x ) F ro m the eq ualities concerning z we get that k ′ = 1 − k . This le ad us to contradiction in the equa lities concerning ¯ z . Ther efore, in this case, if ( X α , Y β , Z γ ) is a collinear triple then there are no α ′ , β ′ , γ ′ such that ( Y β ′ , X α ′ , Z γ ′ ) is a collinea r triple .  In a genera l linear dilata tio n structure the rela tion (2.2.5) do es not hold. Nevertheless, there is some c onten t of this relation which survives in the g eneral context. Prop ositi on 8. 12 F or x, y suﬃciently close d and for ε, µ ∈ Γ with ν ( ε ) , ν ( µ ) ∈ (0 , 1 ) , we have the distanc e estimates: d ( x, w ( x, y , ε, µ )) ≤ ν ( ε ) 1 − ν ( εµ ) d ( x, δ y µ x ) (8.3.7) d ( y , w ( x, y , ε, µ )) ≤ 1 1 − ν ( εµ ) d ( y , δ x ε y ) (8.3.8) 34 Pro of. F urther we shall use the notations fr om the pro of of theorem 8.4, in pa rticular w = w ( x, y , ε, µ ). W e deﬁne by induction four seque nc e s of p oints (the ﬁrst t wo sequences are deﬁned as in rela tion (8.1.1)): x n +1 = δ δ x n ε y n µ x n , y n +1 = δ x n ε y n x ′ n +1 = δ δ y ′ n ε x ′ n µ x n , y ′ n +1 = δ x ′ n +1 ε y ′ n with initial conditions x 0 = x, y 0 = y , x ′ 0 = x, y ′ 0 = δ x ε y . The ﬁrs t tw o seq ue nce s are like in the pr o of of theore m 8 .4, while for the third a nd fourth sequences we have the relations x ′ n = x n , y ′ n = y n +1 . These last sequences come from the fact that they app ear if w e rep ea t the pro o f of theorem 8.4 starting fro m the rela tion: δ δ x ε y µ δ x ε = δ w εµ W e know that all these four seq ue nce s co nverge to w a s n go e s to ∞ . Mor e ov er, we know from the pr o of of theorem 8 .4 that for all n ∈ N we hav e d ( x n , x n +1 ) = d ( x, δ x ε δ y µ x ) ν ( εµ ) n There is an e q uiv alent relation in ter ms of the sequence y ′ n , which is the following: d ( y ′ n , y ′ n +1 ) = d ( δ x ε y , δ δ x ε y µ δ x ε δ x ε y ) ν ( εµ ) n This relation b eco mes: for a ny n ∈ N , n ≥ 1 d ( y n , y n +1 ) = d ( y , δ x ε y ) ν ( εµ ) n +1 F or the ﬁrst dis tance estimate we wr ite: d ( x, w ) ≤ ∞ X n =0 d ( x n , x n +1 ) = d ( x, δ x ε δ y µ x ) ∞ X n =0 ν ( εµ ) n ! = ν ( ε ) 1 − ν ( εµ ) d ( x, δ y µ x ) F or the seco nd distance estimate we write: d ( y , w ) ≤ d ( y , y 1 ) + ∞ X n =1 d ( y n , y n +1 ) = d ( y , y 1 ) + ν ( εµ ) 1 − ν ( εµ ) d ( y , δ x ε y ) = = d ( y , δ x ε y )  1 + ν ( εµ ) 1 − ν ( εµ )  = 1 1 − ν ( εµ ) d ( y , δ x ε y ) and the pro of is done.  F or a collinear triple ( x α , y β , z γ ) in a general linear dilatation structur e we cannot say that x, y , z lie on the same g eo desic. This is fals e, as s hown b y easy exa mples in the Heisenberg group, the simplest noncommutativ e Ca rnot gr oup. Nevertheless, theo rem 8.4 allows to sp eak ab out co llinearity in the sense of deﬁnition 8.10. Aﬃne g eometry is the study of relatio ns which ar e inv ariant with res pe c t to the gro up of aﬃne transfo rmations. An invertible tr a nsformation is a ﬃne if and only if it pr e s erves the ratio of any thr ee co llinea r po ints. W e ar e thus a rriving to the following deﬁnition. 35 Deﬁnition 8.13 L et ( X , d, δ ) b e a line ar dilatation structu r e. A ge ometric al ly aﬃne tr ans- formation T : X → X is a Lipschitz invertible tr ansforma tion su ch that for any c ol line ar triple ( x α , y β , z γ ) the t riple (( T x ) α , ( T y ) β , ( T z ) γ ) is c ol line ar. The group of ge o metric aﬃne tr ansformatio ns deﬁnes a g eometry in the s ense of Er langen progra m. The ma in inv ariants o f such a geometry are collinear triples. There is no obvious connection b etw een collinea rity and geo de s ics of the spa ce, as in classica l aﬃne geometry . (It is worthy to notice that in fact, ther e mig ht b e no geo desics in the metric space ( X , d ) of the linear dila tation structure ( X , d, δ ). F or example, there ar e linear dilata tion structures deﬁned ov er the b oundar y of the dyadic tree [7], whic h is homeomorphic with the middle thirds Cantor set.) The ﬁrst r esult for such a g e o metry is the following. Theorem 8.14 L et ( X , d, δ ) b e a str ong line ar dilatation structu r e. Any Lipschitz, invert- ible, t r ansformation T : ( X , d ) → ( X , d ) is aﬃne in the sense of deﬁnition 7.1 if and only if it is ge ometric al ly aﬃne in the sen s e of deﬁnition 8.13. Pro of. The ﬁrs t implicatio n, namely T aﬃne in the sens e of deﬁnition 7 .1 implies T aﬃne in the sens e of deﬁnition 8.13, is straightforward: by hypothes is on T , for any collinear triple ( x α , y β , z γ ) we ha ve the r elation T δ x α δ y β δ z γ T − 1 = δ T x α δ T y β δ T z γ Therefore, if ( x α , y β , z γ ) is a colline a r triple then the triple (( T x ) α , ( T y ) β , ( T z ) γ ) is collinear . In or de r to show the inv er se implication we use the linear ity o f the dilatatio n structure. Let x, y ∈ X and ε, η ∈ Γ. Then δ x ε δ y η δ x ε − 1 = δ δ x ε y η This identit y lea ds us to the des cription of δ x ε y in ter ms of the ra tio function. 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