Invariance principle and rigidity of high entropy measures
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- by Ali Tahzibi and Jiagang Yang PDF
- Trans. Amer. Math. Soc. 371 (2019), 1231-1251 Request permission
Abstract:
A deep analysis of the Lyapunov exponents for stationary sequence of matrices going back to Furstenberg, for more general linear cocycles by Ledrappier, and generalized to the context of non-linear cocycles by Avila and Viana, gives an invariance principle for invariant measures with vanishing central exponents. In this paper, we give a new criterium formulated in terms of entropy for the invariance principle and, in particular, obtain a simpler proof for some of the known invariance principle results.
As a byproduct, we study ergodic measures of partially hyperbolic diffeomorphisms whose center foliation is one-dimensional and forms a circle bundle. We show that for any such $C^2$ diffeomorphism which is accessible, weak hyperbolicity of ergodic measures of high entropy implies that the system itself is of rotation type.
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Additional Information
- Ali Tahzibi
- Affiliation: Departamento de Matemática, ICMC, Universidade de São Paulo, São Carlos-SP, 13566-590 Brazil
- MR Author ID: 708903
- Email: tahzibi@icmc.usp.br
- Jiagang Yang
- Affiliation: Departamento de Geometria, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, 24020-140 Brazil
- MR Author ID: 949931
- Email: yangjg@impa.br
- Received by editor(s): November 18, 2016
- Received by editor(s) in revised form: May 7, 2017, and May 12, 2017
- Published electronically: October 1, 2018
- Additional Notes: The first author was in a research period at Université Paris-Sud (thanks to support of FAPESP-Brasil:2014/23485-2, CNPq-Brasil) and is thankful for the hospitality of Laboratoire de Topologie and, in particular, Sylvain Crovisier and Jérôme Buzzi for many useful conversations.
The second author was partially supported by CNPq, FAPERJ, and PRONEX - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 1231-1251
- MSC (2010): Primary 37Axx, 37Dxx
- DOI: https://doi.org/10.1090/tran/7278
- MathSciNet review: 3885177