A note on rigidity of Anosov diffeomorphisms of the three torus
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- by F. Micena and A. Tahzibi PDF
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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.References
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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
- MR Author ID: 708903
- Email: tahzibi@icmc.usp.br
- 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
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 2453-2463
- MSC (2010): Primary 37Cxx, 37Dxx
- DOI: https://doi.org/10.1090/proc/14422
- MathSciNet review: 3951424