A note on rigidity of Anosov diffeomorphisms of the three torus

Micena, F.; Tahzibi, A.

doi:10.1090/proc/14422

A note on rigidity of Anosov diffeomorphisms of the three torus
HTML articles powered by AMS MathViewer

by F. Micena and A. Tahzibi PDF

Proc. Amer. Math. Soc. 147 (2019), 2453-2463 Request permission

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

Michael Brin, Dmitri Burago, and Sergey Ivanov, Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus, J. Mod. Dyn. 3 (2009), no. 1, 1–11. MR 2481329, DOI 10.3934/jmd.2009.3.1
John Franks, Anosov diffeomorphisms on tori, Trans. Amer. Math. Soc. 145 (1969), 117–124. MR 253352, DOI 10.1090/S0002-9947-1969-0253352-7
Andrey Gogolev, How typical are pathological foliations in partially hyperbolic dynamics: an example, Israel J. Math. 187 (2012), 493–507. MR 2891713, DOI 10.1007/s11856-011-0088-3
Andrey Gogolev, Bootstrap for local rigidity of Anosov automorphisms on the 3-torus, Comm. Math. Phys. 352 (2017), no. 2, 439–455. MR 3627403, DOI 10.1007/s00220-017-2863-4
Andrey Gogolev and Misha Guysinsky, $C^1$-differentiable conjugacy of Anosov diffeomorphisms on three dimensional torus, Discrete Contin. Dyn. Syst. 22 (2008), no. 1-2, 183–200. MR 2410954, DOI 10.3934/dcds.2008.22.183
Andy Hammerlindl, Leaf conjugacies on the torus, Ergodic Theory Dynam. Systems 33 (2013), no. 3, 896–933. MR 3062906, DOI 10.1017/etds.2012.171
R. de la Llave, Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems, Comm. Math. Phys. 150 (1992), no. 2, 289–320. MR 1194019
F. Ledrappier, Propriétés ergodiques des mesures de Sinaï, Inst. Hautes Études Sci. Publ. Math. 59 (1984), 163–188 (French). MR 743818
Anthony Manning, There are no new Anosov diffeomorphisms on tori, Amer. J. Math. 96 (1974), 422–429. MR 358865, DOI 10.2307/2373551
F. Micena and A. Tahzibi, Regularity of foliations and Lyapunov exponents of partially hyperbolic dynamics on 3-torus, Nonlinearity 26 (2013), no. 4, 1071–1082. MR 3040596, DOI 10.1088/0951-7715/26/4/1071
M. Poletti, Geometric growth for Anosov maps of the three torus, Bull. Brazilian Math. Soc., New Series (2018), https://doi.org/10.1007/s00574-018-0079-7.
Mark Pollicott, Lectures on ergodic theory and Pesin theory on compact manifolds, London Mathematical Society Lecture Note Series, vol. 180, Cambridge University Press, Cambridge, 1993. MR 1215938, DOI 10.1017/CBO9780511752537
V. A. Rohlin, Lectures on the entropy theory of transformations with invariant measure, Uspehi Mat. Nauk 22 (1967), no. 5 (137), 3–56 (Russian). MR 0217258
R. Saghin and J. Yang, Lyapunov exponents and rigidity of Anosov automorphisms and skew products, arXiv 1802.08266, 2018.
Régis Varão, Center foliation: absolute continuity, disintegration and rigidity, Ergodic Theory Dynam. Systems 36 (2016), no. 1, 256–275. MR 3436762, DOI 10.1017/etds.2014.53
Régis Varão, Rigidity for partially hyperbolic diffeomorphisms, Ergodic Theory Dynam. Systems 38 (2018), no. 8, 3188–3200. MR 3868027, DOI 10.1017/etds.2017.11