Let $M$ be an $n$-dimensional manifold supporting a quasi Anosov
diffeomorphism. If $n=3$ then either $M={\mathbb T}^3$, in which case the
diffeomorphisms is Anosov, or else its fundamental group contains a copy of
${\mathbb Z} ^6$. If $n=4$ then $\Pi_1(M)$ contains a copy of ${\mathbb Z} ^4$,
provided that the diffeomorphism is not Anosov.Comment: 5 page

Hertz, Jana Rodriguez

Ures, Raul

Vieitez, Jose L.

English

arXiv

Jana Rodriguez Hertz

Raúl Ures

José L. Vieitez

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 334 (2002) 321–323Systèmes dynamiques/Dynamical SystemsOn manifolds supporting quasi-Anosovdiffeomorphisms *Jana Rodriguez Hertz, Raúl Ures, José L. VieitezCC 30, IMERL – Facultad de Ingeniería, Universidad de la República, Montevideo, UruguayReceived 19 December 2001; accepted 7 January 2002Note presented by Etienne Ghys.Abstract Let M be ann-dimensional manifold supporting a quasi-Anosov diffeomorphism. Ifn = 3then eitherM = T3, in which case the diffeomorphisms is Anosov, or else its fundamentalgroup contains a copy ofZ6. If n = 4 then1(M) contains a copy ofZ4, provided that thediffeomorphism is not Anosov.To cite this article: J. Rodriguez Hertz et al., C. R. Acad.Sci. Paris, Ser. I 334 (2002) 321–323.  2002 Académie des sciences/Éditions scientifiqueset médicales Elsevier SASSur les variétés qui admettent des difféomorphismes de typequasi-AnosovRésumé Soit M une variété différentiable de dimensionn qui admet un difféomorphisme de typequasi-Anosov. Sin = 3 alors on a l’altenative suivante, ou bienM = T3, et dans ce casle difféomorphisme est en fait d’Anosov, ou bien le goupe fondamental deM contientune copie deZ6. Si n = 4, alors 1(M) contient une copie deZ4, pourvu que ledifféomorphisme ne soit pas d’Anosov.Pour citer cet article : J. Rodriguez Hertz et al.,C. R. Acad. Sci. Paris, Ser. I 334 (2002) 321–323.  2002 Académie des sciences/Éditionsscientifiques et médicales Elsevier SAS1. IntroductionIn this work we obtain some restrictions on a manifoldM in order to supportquasi-Anosovdiffeomorphisms (QAD), meaning diffeomorphismsf : M → M such that‖(f n)′(x)v‖ → +∞ for allnon zero vectorsv ∈ TxM, either forn → ∞ or for n → −∞.These maps were introduced in [5] by Mañé, who showed they satisfy Axiom A and the no-cyclecondition. Besides, they are the C1 interior of the class ofexpansive diffeomorphisms g [6], those satisfyingfor someα > 0, that supn∈Z d(f n(x), f n(y)) > α, if x = y.We expect on one hand, that studying the relation between QAD’s (seen as simplified examples ofexpansive diffeomorphisms) and the manifolds supporting them, will bring some ideas about the restrictionsthat expansive homeomorphisms impose on their ambient manifolds. Indeed, let us recall, for instance,that no expansive homeomorphism may be found on the 2-sphere, all expansive homeomorphisms areE-mail addresses: jana@fing.edu.uy (J. Rodriguez Hertz); ures@fing.edu.uy (R. Ures); jvieitez@fing.edu.uy(J.L. Vieitez). 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservésS1631-073X(02)02260-4/FLA 321J. Rodriguez Hertz et al. / C. R. Acad. Sci. Paris, Ser. I 334 (2002) 321–323conjugated to Anosov diffeomorphisms on the 2-torus, and to pseudo-Anosov maps if they live in a surfacewith genus greater than 1 [2,4].In the caseM is a 3-manifold, an expansive C1+ε diffeomorphism is conjugated to an Anosovdiffeomorphism if its non-wandering set isM; in which caseM must beT3 [11].On the other hand, Mañé [7] and Hiraide [3] have posed the question of whether a QAD onTn isnecessarily Anosov. This work provides a positive answer forn = 3, though we must point out that solvingthe problem for higher dimensions seems to involve more sophisticated tools.However, there are also examples of QAD’s on 3-dimensional manifolds having wandering points [1].We present here necessary conditions on a 3-manifold to support this kind of example. Partial results on4-manifolds are also obtained.The main result in this work is the following:THEOREM. – Let f : Mn → Mn be a quasi-Anosov diffeomorphism which is not Anosov. If n = 3 thenπ1(M) contains a subgroup which is isomorphic to Z6. If n = 4, π1(M) contains a subgroup which isisomorphic to Z4.This result is strongly based on a theorem due to R. Plykin, which is stated below, and argumentsconcerning Axiom A diffeomorphisms with the no-cycle condition.THEOREM 1.1 ([8,9]). – Let f : M → M be a diffeomorphism on an n-manifold , where n  3. If f hask expanding attractors or shrinking repellers of codimension 1 then π1(M) contains a subgroup isomorphicto⊕k Zn.We recall that a hyperbolic attractor is said to havedimension u or equivalentlycodimension n − u ifthe dimension of its unstable fibre bundle isu. Analogously we shall say that a hyperbolic repeller is ofdimension s or codimension n − s if the dimension of its stable fiber bundle iss.Our strategy is to see that a QAD on a 3-manifold must have at least two codimension one attractorsor repellers, while on a 4-manifold, it must have at least one. This fact will allow us to prove our maintheorem, getting the following as an immediate corollary:COROLLARY 1.1. – Every quasi-Anosov diffeomorphism on T3 is Anosov.Corollary 1.1 provides an easier way of recognizing Anosov diffeomorphisms onT3: it suffices to checkthat each non zero vector in the tangent bundle goes to infinity under the action ofDf n.2. Relation among dimension 1 attractors and codimension 1 repellersFrom now on, we shall assumeM is a compact smoothn-manifold without boundary, andf : M → Mis a QAD. The non-wandering set off will be denoted by(f ), andi will stand for the basic sets arisingfrom the Spectral Decomposition Theorem [10]. We shall also maintain the standard notationWs(x) forthe stable manifold ofx, i.e., the set of pointsy ∈ M such thatd(f n(x), f n(y)) → 0 asn → ∞.DEFINITION 2.1. – Letting(f ) = 1 ∪ · · · ∪ k , we shall say thati is of type(s, u) if the stabledimension ofi is s, and the unstable dimension ofi is u. The expressionWσ (i) will be used todenote the set⋃x∈i Wσ (x), for σ = s, u, in which case the relationi ≺ j will mean thatWu(i) ∩Ws(j ) = ∅.We recall from [5] that a QAD is Anosov, if it satisfies the strong transversality condition. Notice thatthis condition is satisfied if all basic sets are of the same type. We state this conclusion as a lemmaLEMMA 2.1. –If all basic sets of a QAD f are of the same type then f is Anosov.Therefore we get the following:322To cite this article: J. Rodriguez Hertz et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 321–323PROPOSITION 2.1. – Let f be a QAD such that (f ) = M . If some basic set of f is of type (1, n − 1)(or (n − 1,1)), then f must have a codimension one attractor (repeller).Proof. – Let 1 be a basic set of , of type (1, n − 1) then Wu(1) must meetWs(2) where2is another basic set off (the no cycle condition yields1 = 2). Hence1 ≺ 2, implying that thestable dimension of2 is one. Indeed, letx ∈ 1 andy ∈ 2 such thatz ∈ Ws(x) ∩ Wu(y). If the stabledimension of2 were greater than one, then we would find a non zero vectorv ∈ TzM which would betangent both toWu(z) andWs(z), what would yield‖f n(z)v‖ → 0 as|n| → ∞, contradicting the fact thatf is quasi-Anosov.This leaves us only two possibilities: eitherWu(2) meetsWs(3), for some other basic set3, or 2contains the whole setWu(2), whence it would be a codimension one attractor. In this way we inductivelyobtain a chain1 ≺ 2 ≺ · · ·rwhich must end after a finite number of steps, due to the no cycle condition. Besides, the previous argumentshows thati has stable dimension one, for eachi = 1, . . . , r.If we suppose the previous chain is maximal, thenr must be a codimension one attractor.✷3. Proof of the theoremWe shall see that each QAD on a 3-manifold must have an attractornd a repeller of codimension one,provided that it is not Anosov. This, together with Theorem 1.1 finishes the case dimM = 3.Let f : M3 → M3, since quasi-Anosov maps do not have attracting or repelling periodic points, theirbasic sets must be all of type(1,2) or (2,1). If f has a basic set of type(1,2) then Proposition 2.1 impliesthe existence of a codimension one attractor. Now, iff s not Anosov, then it must also have a basic setof type (2,1), otherwise, all the basic sets would be of the same type. But in this case, Proposition 2.1guarantees the existence of a codimension one repeller, as well.Observe that the previous argument also shows that iff : M4 → M4 is not an Anosov diffeomorphism,then it must have (at least) a basic set of type(1,3) or (3,1), otherwise, all basic sets would be of type(2,2)what would implyf is Anosov. Proposition 2.1 again, implies the existence of an attractor or a repeller ofcodimension one.* The first author was partially supported by a grant from PEDECIBA. The second author was partially supported byCONICYT, Fondo Clemente Estable.References[1] J. Franks, C. Robinson, A quasi-Anosov diffeomorphism that is not Anosov, Trans. Amer. Math. Soc. 223 (1976)267–278.[2] K. Hiraide, Expansive homeomorphisms of surfaces are pseudo-Anosov, Osaka J. Math. 27 (1990) 117–162.[3] K. Hiraide, Dynamical systems of expansive maps on compact manifolds, Sugaku Expo. 5 (1992) 133–154.[4] J. Lewowicz, Expansive homeomorphisms of surfaces, Bol. Soc. Brasil. Mat. 20 (1) (1989) 113–133.[5] R. Mañé, Quasi-Anosov diffeomorphisms and hyperbolic manifolds, Trans. Amer. Math. Soc. 229 (1977) 351–370.[6] R. Mañé, Expansive diffeomorphisms, in: Lecture Notes in Math., Vol. 468, Springer-Verlag, 1975, pp. 162–174.[7] R. Mañé, Personal communication.[8] R. Plykin, Hyperbolic attractors of diffeomorphisms, Russian Math. Surveys 35 (3) (1980) 109–121.[9] R. Plykin, On hyperbolic attractors of diffeomorphisms (the non-orientable case), Russian Math. Surveys 35 (4)(1980) 186–187.[10] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967) 747–817.[11] J. Vieitez, Lyapunov functions and expansive diffeomorphisms on 3D-manifolds, Ergodic Theory Dynam. Systems(to appear).323

On manifolds supporting quasi-Anosov diffeomorphisms

Rodriguez Hertz, Jana

Ures, Raúl

Vieitez, José L.

Numérisation de Documents Anciens Mathématiques

http://www.numdam.org/item/10.1016/S1631-073X(02)02260-4.pdf

On manifolds supporting quasi Anosov diffeomorphisms

On manifolds supporting quasi Anosov diffeomorphisms

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