On Measurable Properties of Anosov Endomorphisms of Torus

TL;DR

This paper investigates the properties of Anosov endomorphisms on the torus, revealing a dichotomy related to the unstable Lyapunov exponent and characterizing absolute continuity of foliations in specific cases.

Contribution

It establishes a dichotomy for the unstable Lyapunov exponent and characterizes absolute continuity of foliations for certain volume-preserving Anosov endomorphisms.

Findings

Either the unstable Lyapunov exponent set has zero measure or the endomorphism is smoothly conjugate to its linearization.

Characterization of absolute continuity of intermediate foliations in volume-preserving cases.

Identification of conditions under which the conjugacy with the linearization is smooth.

Abstract

We found a dichotomy involving the unstable Lyapunov exponent of a special Anosov endomorphism of the torus induced by the conjugacy with the linearization. In fact, either every unstable leaf meets on a set of zero measure the set for which is defined such unstable Lyapunov exponent or the endomorphims is smoothly conjugated with its linearization. Also we are able to characterize the absolute continuity of the intermediate foliation for a class of volume preserving special Anosov endomorphisms of $\mathbb{T}^3.$