Pathological center foliation with dimension greater than one

Abstract

We are considering partially hyperbolic diffeomorphims of the torus, with \begin{document}${\rm{dim}}(E^c) > 1.$\end{document} We prove, under some conditions, that if the all center Lyapunov exponents of the linearization \begin{document}$A,$\end{document} of a DA-diffeomorphism \begin{document}$f,$\end{document} are positive and the center foliation of \begin{document}$f$\end{document} is absolutely continuous, then the sum of the center Lyapunov exponents of \begin{document}$f$\end{document} is bounded by the sum of the center Lyapunov exponents of \begin{document}$A.$\end{document} After, we construct a \begin{document}$C^1-$\end{document} open class of volume preserving DA-diffeomorphisms, far from Anosov diffeomorphisms, with non compact pathological two dimensional center foliation. Indeed, each \begin{document}$f$\end{document} in this open set satisfies the previously established hypothesis, but the sum of the center Lyapunov exponents of \begin{document}$f$\end{document} is greater than the corresponding sum with respect to its linearization. It allows to conclude that the center foliation of \begin{document}$f$\end{document} 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.