Abstract
A dynamical system is called partially hyperbolic if it exhibits three invariant directions, one unstable (expanding), one stable (contracting) and one central direction (somewhere in between the other two). We prove that topologically mixing partially hyperbolic diffeomorphisms whose central direction is non-uniformly contracting (negative Lyapunov exponents) almost everywhere have the Bernoulli property: the system is equivalent to an i. i. d. (independently identically distributed) random process. In particular, these systems are mixing: correlations of integrable functions go to zero as time goes to infinity. We also extend this result in two different ways. Firstly, for 3-dimensional diffeomorphisms, if one requires only non-zero (instead of negative) Lyapunov exponents then one still gets a quasi-Bernoulli property. Secondly, if one assumes accessibility (any two points are joined by some path whose legs are stable segments and unstable segments) then it suffices to requires the mostly contracting property on a positive measure subset, to obtain the same conclusions.
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REFERENCES
F. Abdenur and A. Avila,Robust Transitivity and Topological Mixing for C 1 diffeomorphisms (in preparation).
J. Alves, C. Bonatti, and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding,Inven. Math.140:351–398, 2000.
D. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature,Proc. Steklov. Inst. Math.90, (1967)
A. Arbieto and C.Matheus, The Pasting Lemma I: The vector field casepreprint, 2003.
J. Bochi, B. Fayad, and E. Pujals, As a remark on conservative deffeomorphismspreprint, 2003.
J. Bochi and M. Viana, The Lyapunov Exponents of the generic volume preserving and symplectic systems,preprint, 2002.
C. Bonatti and S. Crovisier, Recurrence and genericite,preprint.
C. Bonatti, L. Díaz, and E. Pujals, AC 1-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources,Annals of Math. 158:355–418, (2003).
C. Bonatti, C.Matheus, M.Viana, and A. Wilkinson, Abundance of stable ergodicity,preprint,www.impa.br/viana,to appear in Commentarii Mathematici Helvetici.
C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting,Israel J. Math.,115:157–193, (2000).
K. Burns, D. Dolgopyat, and Y. Pesin, Partial Hyperbolicity, Lyapunov Exponents and Stable Ergodicity,J. Stat. Phys.,108(5-6):927–942, (2002).
D. Dolgopyat and A. Wilkinson, Stable accessibility isC 1 dense.Asterisque 287:33–60, (2003).
H. Furstenberg, Strictly ergodicity and transformations of the torus,Amer. J. of Math. 83:573–601, (1961).
Y. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory.Russian Mathematical Surveys,32:55–112, (1977).
C. Pugh and M. Shub, Stable Ergodicity and Julienne quasiconformality,J. Eur. Math. Soc. 2:1–52, (2000).
D. Szasz, Boltzmann's ergodic hypothesis, a conjecture for centuries?,Hard ball systems and the Lorentz gas,Encyclopaedia Math. Sci.101:421–448, Springer Verlag, Berlin, (2000).
A. Tahzibi, Robust transitivity implies almost robust ergodicityErg. Th. Dyn. Sys. 24:1261–1269 (2004).
A. Tahzibi, Stably ergodic systems which are not partially hyperbolic.Ph. D. thesis-IMPA, preprint,www.preprint.impa.br/Shadows/SERIE C/2002/9.html, 2002. To appear in Istrael Journal of Mathematics.
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Arbieto, A., Matheus, C. & Pacifico, M.J. The Bernoulli Property for Weakly Hyperbolic Systems. Journal of Statistical Physics 117, 243–260 (2004). https://doi.org/10.1023/B:JOSS.0000044058.99450.c9
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DOI: https://doi.org/10.1023/B:JOSS.0000044058.99450.c9