Hostname: page-component-77c89778f8-fv566 Total loading time: 0 Render date: 2024-07-18T16:18:18.375Z Has data issue: false hasContentIssue false
  • English
  • Français

On the stable ergodicity of Berger–Carrasco’s example

Part of: Smooth dynamical systems: general theory Ergodic theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  27 September 2018

DAVI OBATA
DAVI OBATA*
Affiliation:
CNRS-Laboratoire de Mathématiques d’Orsay, UMR 8628, Université Paris-Sud 11, Orsay Cedex 91405, France Instituto de Matemática, Universidade Federal do Rio de Janeiro, P.O. Box 68530, 21945-970, Rio de Janeiro, Brazil email davi.obata@math.u-psud.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove the stable ergodicity of an example of a volume-preserving, partially hyperbolic diffeomorphism introduced by Berger and Carrasco in [Berger and Carrasco. Non-uniformly hyperbolic diffeomorphisms derived from the standard map. Comm. Math. Phys.329 (2014), 239–262]. This example is robustly non-uniformly hyperbolic, with a two-dimensional center; almost every point has both positive and negative Lyapunov exponents along the center direction and does not admit a dominated splitting of the center direction. The main novelty of our proof is that we do not use accessibility.

Keywords

stable ergodicitynon-uniform hyperbolicitysmooth ergodic theory

MSC classification

Primary: 37A25: Ergodicity, mixing, rates of mixing 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Secondary: 37C40: Smooth ergodic theory, invariant measures
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 4 , April 2020 , pp. 1008 - 1056
Copyright
© Cambridge University Press, 2018

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anosov, D.. Geodesic flows on closed Riemannian manifolds of negative curvature. Proc. Steklov Inst. Math. 90 (1967), iv–235.Google Scholar
Anosov, D. and Sinaĭ, Y.. Certain smooth ergodic systems. Uspekhi Mat. Nauk 22 (1967), 107–172.Google Scholar
Avila, A., Crovisier, S. and Wilkinson, A.. $C^{1}$ density of stable ergodicity. Preprint, 2017, arXiv: 1709.04983.Google Scholar
Barreira, L. and Pesin, Y.. Lyapunov Exponents and Smooth Ergodic Theory (University Lecture Series, 23). American Mathematical Society, Providence, RI, 2002.Google Scholar
Berger, P. and Carrasco, P.. Non-uniformly hyperbolic diffeomorphisms derived from the standard map. Comm. Math. Phys. 329 (2014), 239–262.Google Scholar
Blumenthal, A., Xue, J. and Young, L.. Lyapunov exponents for random perturbations of some area-preserving maps including the standard map. Ann. of Math. (2) 185 (2017), 285–310.Google Scholar
Bonatti, C., Díaz, L. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity (Encyclopaedia of Mathematical Sciences, 102). Springer, Berlin, 2005.Google Scholar
Brin, M. and Pesin, Y.. Partially hyperbolic dynamical systems. Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 170–212.Google Scholar
Brown, A.. Smoothness of stable holonomies inside center-stable manifolds and the C 2 hypothesis in Pugh–Shub and Ledrappier–Young theory. Comment. Math. Helv. 93 (2018), 377–400.Google Scholar
Burns, K., Dolgopyat, D. and Pesin, Y.. Partial hyperbolicity, Lyapunov exponents and stable ergodicity. J. Statist. Phys. 108 (2002), 927–942.Google Scholar
Burns, K. and Wilkinson, A.. On the ergodicity of partially hyperbolic systems. Ann. of Math. (2) 171 (2010), 451–489.Google Scholar
Chirikov, B.. A universal instability of many-dimensional oscillator systems. Phys. Rep. 52 (1979), 264–379.Google Scholar
Crovisier, S. and Pujals, E.. Strongly dissipative surface diffeomorphisms. Comment. Math. Helv. 93 (2018), 377–400.Google Scholar
Duarte, P.. Plenty of elliptic islands for the standard family of area preserving maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), 359–409.Google Scholar
Gorodetski, A.. On stochastic sea of the standard map. Comm. Math. Phys. 309 (2012), 155–192.Google Scholar
Grayson, M., Pugh, C. and Shub, M.. Stably ergodic diffeomorphisms. Ann. of Math. (2) 140 (1994), 295–329.Google Scholar
Hertz, F. R., Hertz, M. R., Tahzibi, A. and Ures, R.. New criteria for ergodicity and non-uniform hyperbolicity. Duke Math. J. 160 (2011), 599–629.Google Scholar
Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, Berlin, 1977.Google Scholar
Hopf, E.. Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung. Ber. Verh. Sächs. Akad. Wiss. Leipzig 91 (1939), 261–304.Google Scholar
Horita, V. and Sambarino, M.. Stable ergodicity and accessibility for certain partially hyperbolic diffeomorphisms with bidimensional center leaves. Comment. Math. Helv. 92 (2017), 467–512.Google Scholar
Izraelev, F.. Nearly linear mappings and their applications. Phys. D 1 (1980), 243–266.Google Scholar
Obata, D.. On the holonomies of strong stable foliations. Notes on D. Obata’s personal web page, 2018.Google Scholar
Pesin, Y.. Characteristic Ljapunov exponents, and smooth ergodic theory. Uspekhi Mat. Nauk 32 (1977), 55–112.Google Scholar
Pugh, C. and Shub, M.. Stable ergodicity and julienne quasi-conformality. J. Eur. Math. Soc. 2 (2000), 1–52.Google Scholar
Pugh, C., Shub, M. and Wilkinson, A.. Hölder foliations. Duke Math. J. 86 (1997), 517–546.Google Scholar
Pugh, C., Shub, M. and Wilkinson, A.. Correction to: ‘Hölder foliations’. Duke Math. J. 105 (2000), 105–106.Google Scholar
Pugh, C., Shub, M. and Wilkinson, A.. Hölder foliations, revisited. J. Mod. Dyn. 6 (2012), 79–120.Google Scholar
Pugh, C., Viana, M. and Wilkinson, A.. Absolute continuity of foliations. M. Viana’s personal web page. Preprint.Google Scholar
Rokhlin, V.. On the Fundamental Ideas of Measure Theory (American Mathematical Society Translation, 71). American Mathematical Society, Providence, RI, 1952.Google Scholar
Shepelyansky, D. and Stone, A.. Chaotic landau level mixing in classical and quantum wells. Phys. Rev. Lett. 74 (1995), 2098–2101.Google Scholar
Sinaĭ, Y.. Topics in Ergodic Theory (Princeton Mathematical Series, 44). Princeton University Press, Princeton, NJ, 1994.Google Scholar
Viana, M.. Multidimensional nonhyperbolic attractors. Publ. Math. Inst. Hautes Études Sci. 85 (1997), 63–96.Google Scholar