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Maximizing measures for partially hyperbolic systems with compact center leaves

Published online by Cambridge University Press:  05 December 2011

F. RODRIGUEZ HERTZ ,
M. A. RODRIGUEZ HERTZ ,
A. TAHZIBI  and
R. URES
F. RODRIGUEZ HERTZ
Affiliation:
IMERL-Facultad de Ingeniería, Universidad de la República, CC 30 Montevideo, Uruguay (email: frhertz@fing.edu.uy, jana@fing.edu.uy, ures@fing.edu.uy)
M. A. RODRIGUEZ HERTZ
Affiliation:
IMERL-Facultad de Ingeniería, Universidad de la República, CC 30 Montevideo, Uruguay (email: frhertz@fing.edu.uy, jana@fing.edu.uy, ures@fing.edu.uy)
A. TAHZIBI
Affiliation:
Departamento de Matemática, ICMC-USP São Carlos, Caixa Postal 668, 13560-970 São, Carlos-SP, Brazil (email: tahzibi@icmc.sc.usp.br)
R. URES
Affiliation:
IMERL-Facultad de Ingeniería, Universidad de la República, CC 30 Montevideo, Uruguay (email: frhertz@fing.edu.uy, jana@fing.edu.uy, ures@fing.edu.uy)
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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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 2: Daniel J. Rudolph – in Memoriam , April 2012 , pp. 825 - 839
Copyright
Copyright © Cambridge University Press 2011

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