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arXiv · 1705.05422

Pathological center foliation with dimension greater than one

J. S. Costa·F. Micena
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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.

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J. S. Costa, F. Micena. 2017-05-15. Pathological center foliation with dimension greater than one. https://arxiv.org/abs/1705.05422

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