Ergodic properties of Viana-like maps with singularities in the base dynamics
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- by José F. Alves and Daniel Schnellmann PDF
- Proc. Amer. Math. Soc. 141 (2013), 3943-3955 Request permission
Abstract:
We consider two examples of Viana maps for which the base dynamics has singularities (discontinuities or critical points) and show the existence of a unique absolutely continuous invariant probability measure and related ergodic properties such as stretched exponential decay of correlations and stretched exponential large deviations.References
- José F. Alves, Strong statistical stability of non-uniformly expanding maps, Nonlinearity 17 (2004), no. 4, 1193–1215. MR 2069701, DOI 10.1088/0951-7715/17/4/004
- José Ferreira Alves and Vítor Araújo, Random perturbations of nonuniformly expanding maps, Astérisque 286 (2003), xvii, 25–62 (English, with English and French summaries). Geometric methods in dynamics. I. MR 2052296
- José F. Alves, Jorge M. Freitas, Stefano Luzzatto, and Sandro Vaienti, From rates of mixing to recurrence times via large deviations, Adv. Math. 228 (2011), no. 2, 1203–1236. MR 2822221, DOI 10.1016/j.aim.2011.06.014
- José F. Alves, Stefano Luzzatto, and Vilton Pinheiro, Markov structures and decay of correlations for non-uniformly expanding dynamical systems, Ann. Inst. H. Poincaré C Anal. Non Linéaire 22 (2005), no. 6, 817–839. MR 2172861, DOI 10.1016/j.anihpc.2004.12.002
- José F. Alves and Marcelo Viana, Statistical stability for robust classes of maps with non-uniform expansion, Ergodic Theory Dynam. Systems 22 (2002), no. 1, 1–32. MR 1889563, DOI 10.1017/S0143385702000019
- V. Araújo and J. Solano, Absolutely continuous invariant measures for non-expanding maps, arXiv:1111.4540v1
- Jérôme Buzzi, Olivier Sester, and Masato Tsujii, Weakly expanding skew-products of quadratic maps, Ergodic Theory Dynam. Systems 23 (2003), no. 5, 1401–1414. MR 2018605, DOI 10.1017/S0143385702001694
- Sébastien Gouëzel, Berry-Esseen theorem and local limit theorem for non uniformly expanding maps, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 6, 997–1024 (English, with English and French summaries). MR 2172207, DOI 10.1016/j.anihpb.2004.09.002
- Sébastien Gouëzel, Decay of correlations for nonuniformly expanding systems, Bull. Soc. Math. France 134 (2006), no. 1, 1–31 (English, with English and French summaries). MR 2233699, DOI 10.24033/bsmf.2500
- Gerhard Keller and Tomasz Nowicki, Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps, Comm. Math. Phys. 149 (1992), no. 1, 31–69. MR 1182410
- Ian Melbourne and Matthew Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems, Comm. Math. Phys. 260 (2005), no. 1, 131–146. MR 2175992, DOI 10.1007/s00220-005-1407-5
- Ian Melbourne and Matthew Nicol, Large deviations for nonuniformly hyperbolic systems, Trans. Amer. Math. Soc. 360 (2008), no. 12, 6661–6676. MR 2434305, DOI 10.1090/S0002-9947-08-04520-0
- Daniel Schnellmann, Non-continuous weakly expanding skew-products of quadratic maps with two positive Lyapunov exponents, Ergodic Theory Dynam. Systems 28 (2008), no. 1, 245–266. MR 2380309, DOI 10.1017/S0143385707000429
- Daniel Schnellmann, Positive Lyapunov exponents for quadratic skew-products over a Misiurewicz-Thurston map, Nonlinearity 22 (2009), no. 11, 2681–2695. MR 2550691, DOI 10.1088/0951-7715/22/11/006
- Marcelo Viana, Multidimensional nonhyperbolic attractors, Inst. Hautes Études Sci. Publ. Math. 85 (1997), 63–96. MR 1471866
- Lai-Sang Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. (2) 147 (1998), no. 3, 585–650. MR 1637655, DOI 10.2307/120960
- Lai-Sang Young, Recurrence times and rates of mixing, Israel J. Math. 110 (1999), 153–188. MR 1750438, DOI 10.1007/BF02808180
Additional Information
- José F. Alves
- Affiliation: Departamento de Matemática, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
- Email: jfalves@fc.up.pt
- Daniel Schnellmann
- Affiliation: Départment de Mathématiques et Applications (DMA), Ecole Normale Supérieure, 45 rue d’Ulm, 75230 Paris cedex 05, France
- Email: daniel.schnellmann@ens.fr
- Received by editor(s): August 22, 2011
- Received by editor(s) in revised form: January 27, 2012
- Published electronically: July 30, 2013
- Additional Notes: The first author was partially supported by Fundação Calouste Gulbenkian, by the European Regional Development Fund through the programme COMPETE and by the Portuguese government through the FCT under the projects PEst-C/MAT/UI0144/2011 and PTDC/MAT/099493/2008
The second author was supported by the Swedish Research Council. - Communicated by: Bryna Kra
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 3943-3955
- MSC (2010): Primary 37A05, 37C40, 37D25; Secondary 60F05, 60F10
- DOI: https://doi.org/10.1090/S0002-9939-2013-11680-1
- MathSciNet review: 3091785