Maximizing measures for partially hyperbolic systems with compact center leaves

By F. R. Hertz, M. R. Hertz, A. Tahzibi, R. Ures. Ergodic Theory and Dynamical Systems, 2010

74 citations recorded by the source corpus.

Abstract We obtain the following dichotomy for accessible partially hyperbolic diffeomorphisms of three-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 are a finite number of entropy-maximizing measures, all of them with non-zero center Lyapunov exponents (at least one with a negative exponent and one with a 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.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b27c63e370e7a7237505ade36f07b0bc28a1f1cd

Citations in the corpus

• Measures of maximal entropy for surface diffeomorphisms (2018, unknown): “Nonuniformly hyperbolic invertible dynamics have been considered: piecewise affine surface homeomorphisms [29], various derived from Anosov [72, 36, 93, 35, 39] or partially hyperbolic systems [80].”
• Accessibility (2018, unknown): “Additionally, it plays a significant role in analyzing the properties of special ergodic measures, as detailed in [HHTU12, VY13, AVW15, CP23].”