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Open sets of Axiom A flows with exponentially mixing attractors


Authors: V. Araújo, O. Butterley and P. Varandas
Journal: Proc. Amer. Math. Soc. 144 (2016), 2971-2984
MSC (2010): Primary 37D20, 37A25; Secondary 37C10
DOI: https://doi.org/10.1090/proc/13055
Published electronically: March 1, 2016
MathSciNet review: 3487229
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Abstract: For any dimension $ d\geq 3$ we construct $ \mathcal {C}^{1}$-open subsets of the space of $ \mathcal {C}^{3}$ vector fields such that the flow associated to each vector field is Axiom A and exhibits a non-trivial attractor which mixes exponentially with respect to the unique SRB measure.


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  • [1] V. Araújo and I. Melbourne,
    Exponential decay of correlations for nonuniformly hyperbolic flows with a $ C^{1+\alpha }$ stable foliation, including the classical Lorenz attractor,
    preprint arXiv:1504.04316, to appear in Annales Henri Poincaré, 2016, .
  • [2] Vítor Araújo and Paulo Varandas, Robust exponential decay of correlations for singular-flows, Comm. Math. Phys. 311 (2012), no. 1, 215-246. MR 2892469, https://doi.org/10.1007/s00220-012-1445-8
  • [3] Artur Avila, Sébastien Gouëzel, and Jean-Christophe Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci. 104 (2006), 143-211. MR 2264836 (2007j:37049), https://doi.org/10.1007/s10240-006-0001-5
  • [4] Viviane Baladi and Brigitte Vallée, Exponential decay of correlations for surface semi-flows without finite Markov partitions, Proc. Amer. Math. Soc. 133 (2005), no. 3, 865-874 (electronic). MR 2113938 (2006d:37047), https://doi.org/10.1090/S0002-9939-04-07671-3
  • [5] Rufus Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95 (1973), 429-460. MR 0339281 (49 #4041)
  • [6] Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989 (56 #1364)
  • [7] O. Butterley and I. Melbourne,
    Disintegration of invariant measures for hyperbolic skew products,
    arXiv:1503.04319, 2015, to appear in Israel J. Math.
  • [8] N. I. Chernov, Markov approximations and decay of correlations for Anosov flows, Ann. of Math. (2) 147 (1998), no. 2, 269-324. MR 1626741 (99d:58101), https://doi.org/10.2307/121010
  • [9] Dmitry Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. (2) 147 (1998), no. 2, 357-390. MR 1626749 (99g:58073), https://doi.org/10.2307/121012
  • [10] Dmitry Dolgopyat, Prevalence of rapid mixing in hyperbolic flows, Ergodic Theory Dynam. Systems 18 (1998), no. 5, 1097-1114. MR 1653299 (2000a:37014), https://doi.org/10.1017/S0143385798117431
  • [11] Dmitry Dolgopyat, Prevalence of rapid mixing. II. Topological prevalence, Ergodic Theory Dynam. Systems 20 (2000), no. 4, 1045-1059. MR 1779392 (2001g:37037), https://doi.org/10.1017/S0143385700000572
  • [12] Michael Field, Ian Melbourne, and Andrei Török, Stability of mixing and rapid mixing for hyperbolic flows, Ann. of Math. (2) 166 (2007), no. 1, 269-291. MR 2342697 (2008i:37053), https://doi.org/10.4007/annals.2007.166.269
  • [13] Boris Hasselblatt and Amie Wilkinson, Prevalence of non-Lipschitz Anosov foliations, Ergodic Theory Dynam. Systems 19 (1999), no. 3, 643-656. MR 1695913 (2000f:37035), https://doi.org/10.1017/S0143385799133868
  • [14] Morris W. Hirsch and Charles C. Pugh, Stable manifolds and hyperbolic sets, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 133-163. MR 0271991 (42 #6872)
  • [15] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173 (58 #18595)
  • [16] Carlangelo Liverani, On contact Anosov flows, Ann. of Math. (2) 159 (2004), no. 3, 1275-1312. MR 2113022 (2005k:37048), https://doi.org/10.4007/annals.2004.159.1275
  • [17] S. Newhouse, D. Ruelle, and F. Takens, Occurrence of strange Axiom A attractors near quasiperiodic flows on $ T^{m}$,$ \,m\geq 3$, Comm. Math. Phys. 64 (1978/79), no. 1, 35-40. MR 516994 (80f:58029)
  • [18] Jacob Palis and Floris Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Studies in Advanced Mathematics, vol. 35, Cambridge University Press, Cambridge, 1993. Fractal dimensions and infinitely many attractors. MR 1237641 (94h:58129)
  • [19] Mark Pollicott, On the rate of mixing of Axiom A flows, Invent. Math. 81 (1985), no. 3, 413-426. MR 807065 (87i:58148), https://doi.org/10.1007/BF01388579
  • [20] Clark Robinson, Dynamical systems, 2nd ed., Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1999. Stability, symbolic dynamics, and chaos. MR 1792240 (2001k:37003)
  • [21] Charles Pugh, Michael Shub, and Amie Wilkinson, Hölder foliations, Duke Math. J. 86 (1997), no. 3, 517-546. MR 1432307 (97m:58155), https://doi.org/10.1215/S0012-7094-97-08616-6
  • [22] David Ruelle, Zeta-functions for expanding maps and Anosov flows, Invent. Math. 34 (1976), no. 3, 231-242. MR 0420720 (54 #8732)
  • [23] David Ruelle, Flots qui ne mélangent pas exponentiellement, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 4, 191-193 (French, with English summary). MR 692974 (84h:58085), https://doi.org/10.1142/9789812833709_0024
  • [24] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. MR 0228014 (37 #3598)
  • [25] R. F. Williams, Expanding attractors, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 169-203. MR 0348794 (50 #1289)
  • [26] Lai-Sang Young, What are SRB measures, and which dynamical systems have them?, J. Statist. Phys. 108 (2002), no. 5-6, 733-754. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. MR 1933431 (2003g:37042), https://doi.org/10.1023/A:1019762724717

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Additional Information

V. Araújo
Affiliation: Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil
Email: vitor.d.araujo@ufba.br, vitor.araujo.im.ufba@gmail.com

O. Butterley
Affiliation: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
Address at time of publication: Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 1-34151 Trieste, Italy
Email: oliver.butterley@univie.ac.at

P. Varandas
Affiliation: Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil
Email: paulo.varandas@ufba.br

DOI: https://doi.org/10.1090/proc/13055
Keywords: Robust exponential decay of correlation, SRB measure, Axiom A flow
Received by editor(s): March 7, 2014
Received by editor(s) in revised form: September 25, 2014, March 19, 2015, and August 28, 2015
Published electronically: March 1, 2016
Additional Notes: The second author is grateful to Henk Bruin for several discussions, and also acknowledges the support of the Austrian Science Fund, Lise Meitner position M1583
The first and third authors were partially supported by CNPq-Brazil, PRONEX-Dyn.Syst. and FAPESB (Brazil).
The authors are deeply grateful to Ian Melbourne for helpful advice and to the anonymous referees for their criticism and many suggestions that helped to improve the article.
Communicated by: Nimish Shah
Article copyright: © Copyright 2016 American Mathematical Society

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