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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Invariance principle and rigidity of high entropy measures


Authors: Ali Tahzibi and Jiagang Yang
Journal: Trans. Amer. Math. Soc. 371 (2019), 1231-1251
MSC (2010): Primary 37Axx, 37Dxx
DOI: https://doi.org/10.1090/tran/7278
Published electronically: October 1, 2018
MathSciNet review: 3885177
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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
Email: tahzibi@icmc.usp.br

Jiagang Yang
Affiliation: Departamento de Geometria, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, 24020-140 Brazil
Email: yangjg@impa.br

DOI: https://doi.org/10.1090/tran/7278
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
Article copyright: © Copyright 2018 American Mathematical Society