Abstract
We prove that -generically, continuum-wise expansive diffeomorphisms satisfy both Axiom A and the no-cycle condition. Moreover, (i) if a volume-preserving diffeomorphism belongs to the -interior of the set of all continuum-wise expansive volume-preserving diffeomorphisms then it is Anosov, and (ii) -generically, every continuum-wise expansive volume-preserving diffeomorphism is transitive Anosov. MSC:37C20, 37D20.
Highlights
Let Diff(M) be the space of diffeomorphisms of closed C∞-manifolds M endowed with the C -topology, and let d denote the distance on M induced from a Riemannian metric· on the tangent bundle TM
We show that C -generically, every continuum-wise expansive diffeomorphism satisfies both Axiom A and the no-cycle condition
We study the continuum-wise expansive case, and if f belongs to the C -interior of CWEμ(M), f is Anosov
Summary
Let Diff(M) be the space of diffeomorphisms of closed C∞-manifolds M endowed with the C -topology, and let d denote the distance on M induced from a Riemannian metric· on the tangent bundle TM. We show that C -generically, every continuum-wise expansive diffeomorphism satisfies both Axiom A and the no-cycle condition. Theorem A For C -generic f , if f is continuum-wise expansive f satisfies both Axiom A and the no-cycle condition. Let CWEμ(M) be the set of all continuum-wise expansive volume-preserving diffeomorphisms.
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