Continuum-wise expansiveness for generic diffeomorphisms

TL;DR

This paper investigates the properties of continuum-wise expansive diffeomorphisms on closed manifolds, showing that generically they satisfy Axiom A without cycles, but not all partially hyperbolic diffeomorphisms are continuum-wise expansive.

Contribution

It establishes that generically continuum-wise expansive diffeomorphisms satisfy Axiom A without cycles and provides a counterexample among partially hyperbolic diffeomorphisms.

Findings

Generic continuum-wise expansive diffeomorphisms satisfy Axiom A without cycles

Existence of partially hyperbolic diffeomorphisms that are not continuum-wise expansive

Abstract

Let $M$ be a closed smooth manifold and let $f:M\to M$ be a diffeomorphism. $C^1$-generically, a continuum-wise expansive satisfies Axiom A without cycles. Moreover, there is a partially hyperbolic diffeomorphism $f$ such that it is not continuum-wise expansive.