# Pathological center foliation with dimension greater than one

## TL;DR

This paper investigates the properties of partially hyperbolic diffeomorphisms on the torus with higher-dimensional center foliations, establishing bounds on Lyapunov exponents and constructing examples with pathological center foliations.

## Contribution

It proves bounds on Lyapunov exponents for such diffeomorphisms and constructs new examples with non-absolutely continuous and pathological center foliations.

## Key findings

- Bound on sum of center Lyapunov exponents for certain diffeomorphisms.
- Existence of open class of volume-preserving diffeomorphisms with non-absolute continuous center foliation.
- Construction of examples with disintegration of volume neither Lebesgue nor atomic.

## Abstract

In this paper we are considering partially hyperbolic diffeomorphims of the torus, with $dim(E^c) > 1.$ We prove, under some conditions, that if the all center Lyapunov exponents of the linearization $A,$ of a \mbox{DA-diffeomorphism} $f,$ are positive and the center foliation of $f$ is absolutely continuous, then the sum of the center Lyapunov exponents of $f$ is bounded by the sum of the center Lyapunov exponents of $A.$ After, we construct a $C^1-$open class of volume preserving \mbox{DA-diffeomorphisms}, far from Anosov diffeomorphisms, with non compact pathological two dimensional center foliation. Indeed, each $f$ in this open set satisfies the previously established hypothesis, but the sum of the center Lyapunov exponents of $f$ is greater than the corresponding sum with respect to its linearization. It allows to conclude that the center foliation of $f$ is non absolutely continuous. We still build an example of a DA-diffeomorphism, such that the disintegration of volume along the two dimensional, non compact center foliation is neither Lebesgue nor atomic.