Realization of Anosov Diffeomorphisms on the Torus

TL;DR

This paper explores the pressure functions of area-preserving Anosov maps on the torus, showing their equivalence to linear automorphisms and addressing a conjecture about their conjugacy.

Contribution

It demonstrates the equality of pressure functions for Anosov diffeomorphisms and linear automorphisms, providing a negative answer to a conjecture on smooth conjugacy.

Findings

Pressure functions for Anosov maps match those of linear automorphisms.

Counterexample to the conjecture on $C^{1+eta}$ conjugacy based on pressure functions.

Negative resolution to the question on geometric potentials and conjugacy.

Abstract

We study area preserving Anosov maps on the two-dimensional torus within a fixed homotopy class. We show that the set of pressure functions for Anosov diffeomorphisms with respect to the geometric potential is equal to the set of pressure functions for the linear Anosov automorphism with respect to H\"{o}lder potentials. We use this result to provide a negative answer to the $C^{1+\alpha}$ version of the question posed by Rodriguez Hertz on whether two homotopic area preserving $C^\infty$ Anosov difeomorphisms whose geometric potentials have identical pressure functions must be $C^\infty$ conjugate.