Diffeomorphisms Holder conjugate to Anosov diffeomorphisms

TL;DR

The paper presents a counterexample showing that a Holder conjugate of an Anosov diffeomorphism need not be Anosov, but also provides conditions under which conjugacy implies the same dynamical properties.

Contribution

It constructs a counterexample for $C^{1+Lip}$ conjugates and establishes conditions for conjugacy to imply Anosov behavior.

Findings

Counterexample of non-Anosov conjugate with high smoothness

Conditions under which Holder conjugacy implies Anosov property

Extension of Fisher's 2006 result to higher smoothness classes

Abstract

We show by means of a counterexample that a $C^{1+Lip}$ diffeomorphism Holder conjugate to an Anosov diffeomorphism is not necessarily Anosov. The counterexample can bear higher smoothness up to $C^3$. Also we include a result from the 2006 Ph.D. thesis of T. Fisher: a $C^{1+Lip}$ diffeomorphism Holder conjugate to an Anosov diffeomorphism is Anosov itself provided that Holder exponents of the conjugacy and its inverse are sufficiently large.