Ergodic properties of Viana-like maps with singularities in the base dynamics

Authors:
José F. Alves and Daniel Schnellmann

Journal:
Proc. Amer. Math. Soc. **141** (2013), 3943-3955

MSC (2010):
Primary 37A05, 37C40, 37D25; Secondary 60F05, 60F10

Published electronically:
July 30, 2013

MathSciNet review:
3091785

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Abstract | References | Similar Articles | Additional Information

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.

**[A]**José F. Alves,*Strong statistical stability of non-uniformly expanding maps*, Nonlinearity**17**(2004), no. 4, 1193–1215. MR**2069701**, 10.1088/0951-7715/17/4/004**[AA]**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****[AFLV]**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**, 10.1016/j.aim.2011.06.014**[ALP]**José F. Alves, Stefano Luzzatto, and Vilton Pinheiro,*Markov structures and decay of correlations for non-uniformly expanding dynamical systems*, Ann. Inst. H. Poincaré Anal. Non Linéaire**22**(2005), no. 6, 817–839. MR**2172861**, 10.1016/j.anihpc.2004.12.002**[AV]**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**, 10.1017/S0143385702000019**[AS]**V. Araújo and J. Solano,*Absolutely continuous invariant measures for non-expanding maps*, arXiv:1111.4540v1**[BST]**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**, 10.1017/S0143385702001694**[G1]**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**, 10.1016/j.anihpb.2004.09.002**[G2]**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****[KN]**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****[MN1]**Ian Melbourne and Matthew Nicol,*Almost sure invariance principle for nonuniformly hyperbolic systems*, Comm. Math. Phys.**260**(2005), no. 1, 131–146. MR**2175992**, 10.1007/s00220-005-1407-5**[MN2]**Ian Melbourne and Matthew Nicol,*Large deviations for nonuniformly hyperbolic systems*, Trans. Amer. Math. Soc.**360**(2008), no. 12, 6661–6676. MR**2434305**, 10.1090/S0002-9947-08-04520-0**[S1]**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**, 10.1017/S0143385707000429**[S2]**Daniel Schnellmann,*Positive Lyapunov exponents for quadratic skew-products over a Misiurewicz-Thurston map*, Nonlinearity**22**(2009), no. 11, 2681–2695. MR**2550691**, 10.1088/0951-7715/22/11/006**[V]**Marcelo Viana,*Multidimensional nonhyperbolic attractors*, Inst. Hautes Études Sci. Publ. Math.**85**(1997), 63–96. MR**1471866****[Y1]**Lai-Sang Young,*Statistical properties of dynamical systems with some hyperbolicity*, Ann. of Math. (2)**147**(1998), no. 3, 585–650. MR**1637655**, 10.2307/120960**[Y2]**Lai-Sang Young,*Recurrence times and rates of mixing*, Israel J. Math.**110**(1999), 153–188. MR**1750438**, 10.1007/BF02808180

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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

DOI:
https://doi.org/10.1090/S0002-9939-2013-11680-1

Keywords:
Almost Sure Invariance Principle,
Berry-Esseen Theorem,
Central Limit Theorem,
decay of correlations,
large deviations,
Local Limit Theorem,
Viana maps

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

Article copyright:
© Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.