On Lyapunov exponents properties of special Anosov endomorphisms on ${\mathbb{T}}^d$

Abstract

This work is addressed to study Anosov endomorphisms of ${\mathbb{T}}^d,$ $d\ge 3.$ We are interested to obtain metric and topological information on such Anosov endomorphism by comparison between their Lyapunov exponents and the ones of its linearization. We can characterize when a weak unstable foliation of a special Anosov endomorphism near to linear is an absolutely continuous foliation. Also, we show that in dimension $d\ge 3,$ it is possible to find a smooth special Anosov endomorphism being conservative but not Lipschitz conjugate with its linearization, in contrast with the smooth rigidity in dimension two.