Markov structures for non-uniformly expanding maps on compact manifolds in arbitrary dimension
Authors:
José F. Alves, Stefano Luzzatto and Vilton Pinheiro
Journal:
Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 26-31
MSC (2000):
Primary 37D20, 37D50, 37C40
DOI:
https://doi.org/10.1090/S1079-6762-03-00106-9
Published electronically:
February 14, 2003
MathSciNet review:
1988869
Full-text PDF Free Access
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Abstract: We consider non-uniformly expanding maps on compact Riemannian manifolds of arbitrary dimension, possibly having discontinuities and/or critical sets, and show that under some general conditions they admit an induced Markov tower structure for which the decay of the tail of the return time function can be controlled in terms of the time generic points needed to achieve some uniform expanding behavior. As a consequence we obtain some rates for the decay of correlations of those maps and conditions for the validity of the Central Limit Theorem.
- José Ferreira Alves, SRB measures for non-hyperbolic systems with multidimensional expansion, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 1, 1–32 (English, with English and French summaries). MR 1743717, DOI 10.1016/S0012-9593(00)00101-4
- 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
ALP1dim J. F. Alves, S. Luzzatto, V. Pinheiro, Lyapunov exponent and rates of mixing for one-dimensional maps, Preprint 2002.
ALPJ. F. Alves, S. Luzzatto, V. Pinheiro, Markov structures and decay of correlations for non-uniformly expanding dynamical systems, Preprint 2002.
AV J. F. Alves, M. Viana, Statistical stability for robust classes of maps with non-uniform expansion, Ergod. Th. & Dynam. Sys. 22 (2002), 1-32.
- Marcelo Viana, Homoclinic bifurcations and persistence of nonuniformly hyperbolic attractors, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 1221–1229. MR 1404023
- 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
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Al J. F. Alves, SRB measures for non-hyperbolic systems with multidimensional expansion, Ann. Scient. Éc. Norm. Sup., $4^e$ série, 33 (2000), 1-32.
ABV J. F. Alves, C. Bonatti, M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math. 140 (2000), 351-398.
ALP1dim J. F. Alves, S. Luzzatto, V. Pinheiro, Lyapunov exponent and rates of mixing for one-dimensional maps, Preprint 2002.
ALPJ. F. Alves, S. Luzzatto, V. Pinheiro, Markov structures and decay of correlations for non-uniformly expanding dynamical systems, Preprint 2002.
AV J. F. Alves, M. Viana, Statistical stability for robust classes of maps with non-uniform expansion, Ergod. Th. & Dynam. Sys. 22 (2002), 1-32.
V M. Viana, Multidimensional non-hyperbolic attractors, Publ. Math. IHES 85 (1997), 63-96.
Y1 L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math. 147 (1998), 585-650.
Y2 L.-S. Young, Recurrence times and rates of mixing, Israel J. Math. 110 (1999), 153-188.
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Additional Information
José F. Alves
Affiliation:
Departamento de Matemática Pura, Faculdade de Ciências do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
Email:
jfalves@fc.up.pt
Stefano Luzzatto
Affiliation:
Mathematics Department, Imperial College, 180 Queen’s Gate, London SW7, UK
Email:
stefano.luzzatto@ic.ac.uk
Vilton Pinheiro
Affiliation:
Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil
Email:
viltonj@ufba.br
Received by editor(s):
November 5, 2002
Published electronically:
February 14, 2003
Additional Notes:
Work carried out at the Federal University of Bahia, University of Porto and Imperial College, London. Partially supported by CMUP, PRODYN, SAPIENS and UFBA
Communicated by:
Svetlana Katok
Article copyright:
© Copyright 2003
American Mathematical Society
