A quasi-Anosov diffeomorphism that is not Anosov

Abstract

ABSTRACT. In this note, we give an example of a diffeomorphism / on a three dimensional manifold M such that /has a property called quasi-Anosov but such that / does not have a hyperbolic structure (is not Anosov). Mané has given a method of extending / to a diffeomorphism g on a larger dimensional manifold V such that g has a hyperbolic structure on M as a subset of V. This gives a counterexample to a question of M. Hirsch.M. Hirsch asks in [2], if a diffeomorphism g: V -► V has a compact invariant submanifold M C V with a hyperbolic structure as a subset of V, does it folfollow that g restricted to M is Anosov (has a hyperbolic structure). He proves this is true in certain cases if g has a dense orbit in V. Ricardo MarTé notes that g restricted to M has a property he calls quasi-Anosov [5]. He asks if a quasiAnosov diffeomorphism is always Anosov. C. Robinson [6] gives an example of a quasi-Anosov flow (not a diffeomorphism) that is not Anosov on an eleven dimensional manifold. In this note, we give an example of a quasi-Anosov diffeomorphism / on a three dimensional manifold. (This is the minimal dimension.) Mane gives a method in [5] of embedding our result in a diffeomorphism g of a manifold V such that g has a hyperbolic structure on M. This gives a counterexample to the question of Hirsch as stated above. However, we do not know if g can be constructed so M is contained in the nonwandering set of g. Also, the results of Hirsch [2] show that if our f:M3 -*■ M3 is embedded in g: V-► V so that g has a hyperbolic structure on M and the dimension of Kis four or five, then g cannot have a point with a dense orbit in all of V.1. Definitions and Theorem. A diffeomorphism is quasi-Anosov if the fact that | Tf"v\ is bounded for all n £ Z implies that v = 0. Here Tf is the induced map on tangent vectors of M, v E TM. If A C M is invariant by a diffeomorphism /, we say that /has a hyperbolic structure on A if there are 0 < X < 1, C> 0, and a splitting TM IA = F" 0 Es such that for n > 0,

Quasi-Anosov diffeomorphisms of 3-manifolds

“…The image of D under this second conjugacy consists of two spheres S 3 , each bounding a ball containing A in its interior. So, the fact that D is a fundamental domain implies that M is the connected sum of two tori, just like in the Franks-Robinson example [3].…”

“…It is related to a problem posed by M. Hirsch, around 1969: given a diffeomorphism f : N → N and a compact invariant hyperbolic set Λ of f , describe the topology of Λ and the dynamics of f restricted to Λ. Hirsch asked, in particular, whether the fact that Λ were a manifold M would imply that the restriction of f to M is an Anosov diffeomorphism [11]. However, in 1976, Franks and Robinson gave an example of non-Anosov hyperbolic sub-dynamics in the connected sum of two T 3 [3] (see below). There are also examples of hyperbolic sub-dynamics in non-orientable 3-manifolds; for instance, the example of Zhuzhoma and Medvedev [18].…”

On structurally stable diffeomorphisms with codimension one expanding attractors

“…Moreover, any such surface admits a structurally stable diffeomorphism with orientable pseudotame basic sets. As to closed n-manifolds for n ≥ 3, the first example of a structurally stable DA-diffeomorphism of T 3 with a codimension one orientable expanding attractor was constructed carefully by Franks and Robinson [13]. This construction was generalized by Plykin [45] for n ≥ 3 in the framework of the topological classification of codimension one pseudotame basic sets.…”

“…This construction was generalized by Plykin [45] for n ≥ 3 in the framework of the topological classification of codimension one pseudotame basic sets. In [13], one constructs a diffeomorphism f of the connected sum T 3 T 3 that is quasi-Anosov but not Anosov. Actually, f has two orientable codimension one pseudotame basic sets, one of them an expanding attractor and the other a contracting repeller.…”

“…Actually, f is constructed by the 'surgery operation' described by Smale [53] and Williams [59] to get a DA-diffeomorphism with a one-dimensional basic set on the 2-torus. The neat construction is in [31] and [48] (for the 2-torus), and [13] for the 3-torus, and [45] (for any n-torus, n ≥ 2). According to Theorem 2.3 in [45], there is a structurally stable DA-diffeomorphism f : T n → T n having a unique expanding orientable codimension one attractor Ω and the data set {O j , 1} r j=1 .…”