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arXiv · 1010.3372

Maximizing measures for partially hyperbolic systems with compact center leaves

F. Rodriguez Hertz·M. A. Rodriguez Hertz·A. Tahzibi·R. Ures
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Abstract

We obtain the following dichotomy for accessible partially hyperbolic diffeomorphisms of 3-dimensional manifolds having compact center leaves: either there is a unique entropy maximizing measure, this measure has the Bernoulli property and its center Lyapunov exponent is 0 or, there is a finite number of entropy maximizing measures, all of them with nonzero center Lyapunov exponent (at least one with negative exponent and one with positive exponent), that are finite extensions of a Bernoulli system. In the first case of the dichotomy we obtain that the system is topologically conjugated to a rotation extension of a hyperbolic system. This implies that the second case of the dichotomy holds for an open and dense set of diffeomorphisms in the hypothesis of our result. As a consequence we obtain an open set of topologically mixing diffeomorphisms having more than one entropy maximizing measure.

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F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi, R. Ures. 2010-10-16. Maximizing measures for partially hyperbolic systems with compact center leaves. https://arxiv.org/abs/1010.3372

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