2017/05/15 by Costa, J. S., Micena, F.
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1705.05422
In this paper we are considering partially hyperbolic diffeomorphims of the torus, with dim(Ec) > 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 C1-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.