New Derived from Anosov Diffeomorphisms with pathological center foliation

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 $f:{\mathbb{T}}^3\to {\mathbb{T}}^3$ is a volume preserving DA diffeomorphism homotopic to linear Anosov $A,$ such that the center Lyapunov exponent satisfies ${\lambda }_f^c(x)>{\lambda }_A^c>0,$ 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 ${\lambda }_f^c(x)>{\lambda }_A^c>0$ for $m-$almost everywhere $x\in {\mathbb{T}}^3.$ Particularly for every $f\in U,$ the center foliation of $f$ is non absolutely continuous.

References in corpus (1)

• A non-dynamically coherent example on ${\mathbb{T}}^3$