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

Abstract

We consider a smooth area-preserving Anosov diffeomorphism $f:{\mathbb{T}}^2\to {\mathbb{T}}^2$ 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.

v2: structural changes to exposition; 27 pages, 7 figures. Comments are welcome