New Derived from Anosov Diffeomorphisms with pathological center foliation

TL;DR

This paper investigates derived from Anosov diffeomorphisms on the 3-torus, demonstrating that under certain conditions, their center foliations are non-absolutely continuous, and introduces a new class of such diffeomorphisms.

Contribution

It constructs a new open class of volume-preserving DA diffeomorphisms with pathological center foliation behavior on the 3-torus.

Findings

Center foliation is non absolutely continuous under specified conditions.

Constructs a new class of non-Anosov volume-preserving DA diffeomorphisms.

Center Lyapunov exponent exceeds that of the linear automorphism.

Abstract

In this paper we focused our study on Derived From Anosov diffeomorphisms (DA diffeomorphisms ) of the torus $\mathbb{T}^3,$ it is, an absolute partially hyperbolic diffeomorphism on $\mathbb{T}^3$ homotopic to an Anosov linear automorphism of the $\mathbb{T}^3.$ We can prove that if $f: \mathbb{T}^3 \rightarrow \mathbb{T}^3 $ is a volume preserving DA diffeomorphism homotopic to linear Anosov $A,$ such that the center Lyapunov exponent satisfies $\lambda^c_f(x) > \lambda^c_A > 0,$ with $x $ belongs to a positive volume set, then the center foliation of $f$ is non absolutely continuous. We construct a new open class $U$ of non Anosov and volume preserving DA diffeomorphisms, satisfying the property $\lambda^c_f(x) > \lambda^c_A > 0$ for $m-$almost everywhere $x \in \mathbb{T}^3.$ Particularly for every $f \in U,$ the center foliation of $f$ is non absolutely continuous.