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

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