Flexibility of Lyapunov exponents with respect to two classes of measures on the torus

TL;DR

This paper investigates the constraints on Lyapunov exponents and topological entropy for smooth area-preserving Anosov diffeomorphisms on the torus, revealing that the known inequalities are the only restrictions on these invariants.

Contribution

It demonstrates that the established inequalities between Lyapunov exponents and topological entropy are the only possible restrictions for such dynamical systems.

Findings

The inequalities between Lyapunov exponents and entropy are the only restrictions.

Equalities occur only under specific conditions.

The results clarify the relationship between measure-theoretic and topological invariants.

Abstract

We consider a smooth area-preserving Anosov diffeomorphism $f\colon \mathbb T^2\rightarrow \mathbb T^2$ homotopic to an Anosov automorphism $L$ of $\mathbb T^2$. It is known that the positive Lyapunov exponent of $f$ with respect to the normalized Lebesgue measure is less than or equal to the topological entropy of $L$, which, in addition, is less than or equal to the Lyapunov exponent of $f$ with respect to the probability measure of maximal entropy. Moreover, the equalities only occur simultaneously. We show that these are the only restrictions on these two dynamical invariants.