The Teichm\"uller space of an Anosov diffeomorphism of $T^2$

TL;DR

This paper characterizes the space of smooth conjugacy classes of Anosov diffeomorphisms on the 2-torus, showing that all pairs of cohomology classes related to eigenvalues occur within these classes.

Contribution

It demonstrates that every pair of Hölder reduced cohomology classes can be realized by an Anosov diffeomorphism on the 2-torus, answering a fundamental question about their spectral invariants.

Findings

All pairs of Hölder reduced cohomology classes occur as eigenvalue invariants.

The space of conjugacy classes is fully characterized by these cohomology pairs.

The result applies specifically to the 2-torus, the only surface supporting Anosov diffeomorphisms.

Abstract

In this paper we consider the space of smooth conjugacy classes of an Anosov diffeomorphism of the two-torus. The only 2-manifold that supports an Anosov diffeomorphism is the 2-torus, and Franks and Manning showed that every such diffeomorphism is topologically conjugate to a linear example, and furthermore, the eigenvalues at periodic points are a complete smooth invariant. The question arises: what sets of eigenvalues occur as the Anosov diffeomorphism ranges over a topological conjugacy class? This question can be reformulated: what pairs of cohomology classes (one determined by the expanding eigenvalues, and one by the contracting eigenvalues) occur as the diffeomorphism ranges over a topological conjugacy class? The purpose of this paper is to answer this question: all pairs of H\"{o}lder reduced cohomology classes occur.