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Pathological center foliation with dimension greater than one

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  • We are considering partially hyperbolic diffeomorphims of the torus, with ${\rm{dim}}(E^c) > 1.$ We prove, under some conditions, that if the all center Lyapunov exponents of the linearization $A,$ of a 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 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.

    Mathematics Subject Classification: Primary: 37D10, 37D20, 37D25, 37D30.

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  • Figure 1.   

    Figure 2.  The fixed point $p_n$ of $g_n$ and $g_{n,j}$, respectively

    Figure 3.   

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