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A note on rigidity of Anosov diffeomorphisms of the three torus


Authors: F. Micena and A. Tahzibi
Journal: Proc. Amer. Math. Soc. 147 (2019), 2453-2463
MSC (2010): Primary 37Cxx, 37Dxx
DOI: https://doi.org/10.1090/proc/14422
Published electronically: February 20, 2019
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Abstract: We consider Anosov diffeomorphisms on $ \mathbb{T}^3$ such that the tangent bundle splits into three subbundles $ E^s_f \oplus E^{wu}_f \oplus E^{su}_f.$ We show that if $ f$ is $ C^r, r \geq 2,$ volume preserving, then $ f$ is $ C^1$ conjugated with its linear part $ A$ if and only if the center foliation $ \mathcal {F}^{wu}_f$ is absolutely continuous and the equality $ \lambda ^{wu}_f(x) = \lambda ^{wu}_A,$ between center Lyapunov exponents of $ f$ and $ A,$ holds for $ m$ a.e. $ x \in \mathbb{T}^3.$ We also conclude rigidity derived from Anosov diffeomorphism, assuming a strong absolute continuity property (Uniform Bounded Density property) of strong stable and strong unstable foliations.


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Additional Information

F. Micena
Affiliation: IMC, Universidade Federal de Itajuba, Itajubá-MG, 37500-903, Brazil
Email: fpmicena82@unifei.edu.br

A. Tahzibi
Affiliation: ICMC, Universidade de Sao Paulo, São Carlos-SP, 13566-590, Brazil
Email: tahzibi@icmc.usp.br

DOI: https://doi.org/10.1090/proc/14422
Received by editor(s): May 21, 2018
Received by editor(s) in revised form: June 27, 2018
Published electronically: February 20, 2019
Communicated by: Nimish Shah
Article copyright: © Copyright 2019 American Mathematical Society