On the uniform hyperbolicity of some nonuniformly hyperbolic systems
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- by José F. Alves, Vítor Araújo and Benoît Saussol PDF
- Proc. Amer. Math. Soc. 131 (2003), 1303-1309 Request permission
Abstract:
We give sufficient conditions for the uniform hyperbolicity of certain nonuniformly hyperbolic dynamical systems. In particular, we show that local diffeomorphisms that are nonuniformly expanding on sets of total probability (probability one with respect to every invariant probability measure) are necessarily uniformly expanding. We also present a version of this result for diffeomorphisms with nonuniformly hyperbolic sets.References
- J. F. Alves, V. Araújo, Random perturbations of nonuniformly expanding maps, to appear in Astérisque.
- J. F. Alves, V. Araújo, Random perturbations of partially hyperbolic systems, in preparation.
- Christian Bonatti and Marcelo Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math. 115 (2000), 157–193. MR 1749677, DOI 10.1007/BF02810585
- Christian Bonatti and Marcelo Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math. 115 (2000), 157–193. MR 1749677, DOI 10.1007/BF02810585
- Henk Bruin and Gerhard Keller, Equilibrium states for $S$-unimodal maps, Ergodic Theory Dynam. Systems 18 (1998), no. 4, 765–789. MR 1645373, DOI 10.1017/S0143385798108337
- Franz Hofbauer and Gerhard Keller, Quadratic maps without asymptotic measure, Comm. Math. Phys. 127 (1990), no. 2, 319–337. MR 1037108
- Yuri Kifer, General random perturbations of hyperbolic and expanding transformations, J. Analyse Math. 47 (1986), 111–150. MR 874047, DOI 10.1007/BF02792535
- K. Krzyżewski and W. Szlenk, On invariant measures for expanding differentiable mappings, Studia Math. 33 (1969), 83–92. MR 245761, DOI 10.4064/sm-33-1-83-92
- Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
- V. A. Pliss, On a conjecture of Smale, Differencial′nye Uravnenija 8 (1972), 268–282 (Russian). MR 0299909
- David Ruelle, The thermodynamic formalism for expanding maps, Comm. Math. Phys. 125 (1989), no. 2, 239–262. MR 1016871
- Michael Shub, Global stability of dynamical systems, Springer-Verlag, New York, 1987. With the collaboration of Albert Fathi and Rémi Langevin; Translated from the French by Joseph Christy. MR 869255, DOI 10.1007/978-1-4757-1947-5
- Michael Shub and Amie Wilkinson, Pathological foliations and removable zero exponents, Invent. Math. 139 (2000), no. 3, 495–508. MR 1738057, DOI 10.1007/s002229900035
- A. Tahzibi, Stably ergodic systems that are not partially hyperbolic, Preprint IMPA 2001.
Additional Information
- José F. Alves
- Affiliation: Centro de Matemática da Universidade do Porto, 4169-007 Porto, Portugal
- Email: jfalves@fc.up.pt
- Vítor Araújo
- Affiliation: Centro de Matemática da Universidade do Porto, 4169-007 Porto, Portugal
- MR Author ID: 665394
- Email: vdaraujo@fc.up.pt
- Benoît Saussol
- Affiliation: LAMFA - Université de Picardie Jules Verne, 33 rue St Leu, 80039 Amiens, France
- Email: benoit.saussol@u-picardie.fr
- Received by editor(s): November 21, 2001
- Published electronically: November 20, 2002
- Additional Notes: The authors were partially supported by PRODYN
- Communicated by: Michael Handel
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 1303-1309
- MSC (2000): Primary 58F15, 58F99
- DOI: https://doi.org/10.1090/S0002-9939-02-06857-0
- MathSciNet review: 1948124