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Volume lemmas and large deviations for partially hyperbolic endomorphisms

Part of: Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  24 September 2019

ANDERSON CRUZ ,
GIOVANE FERREIRA  and
PAULO VARANDAS Open the ORCID record for PAULO VARANDAS [Opens in a new window]
ANDERSON CRUZ
Affiliation:
Centro de Ciências Exatas e Tecnológicas, Universidade Federal do Recôncavo da Bahia, Av. Rui Barbosa s/n, 44380-000Cruz das Almas, BA, Brazil email anderson.cruz@ufrb.edu.br
GIOVANE FERREIRA
Affiliation:
Departamento de Matemática, Universidade Federal do Maranhão, Av. dos Portugueses 1966, Vila Bacanga, 65065-545São Luís, MA, Brazil email giovane.ferreira@ufma.br
PAULO VARANDAS
Affiliation:
Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110Salvador, Brazil email paulo.varandas@ufba.br
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Abstract

We consider partially hyperbolic attractors for non-singular endomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. We prove volume lemmas for both Lebesgue measure on the topological basin of the attractor and the SRB measure supported on the attractor. As a consequence, under a mild assumption we prove exponential large-deviation bounds for the convergence of Birkhoff averages associated to continuous observables with respect to the SRB measure.

Keywords

volume lemmaspartially hyperbolic endomorphismSRB measurelarge deviations

MSC classification

Primary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Secondary: 37D30: Partially hyperbolic systems and dominated splittings 37D35: Thermodynamic formalism, variational principles, equilibrium states
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 1 , January 2021 , pp. 213 - 240
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Copyright
© Cambridge University Press, 2019

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References

Alves, J. F., Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140(2) (2000), 351–398.Google Scholar
Araujo, V. and Pacifico, M.. Large deviations for non-uniformly expanding maps. J. Stat. Phys. 25 (2006), 411–453 (revised version arXiv:0601449v4).Google Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin, 1975.Google Scholar
Cao, Y., Feng, D. and Huang, W.. The thermodynamic formalism for sub-additive potentials. Discrete Contin. Dyn. Syst. 20 (2008), 639–657.Google Scholar
Carvalho, M.. Sinai–Ruelle–Bowen measures for N-dimensional derived from Anosov diffeomorphisms. Ergod. Th. & Dynam. Sys. 13(1) (1993), 21–44.Google Scholar
Chung, Y.-M. and Takahasi, H.. Large deviation principle for Benedicks–Carleson quadratic maps. Comm. Math. Phys. 315 (2012), 803–826.Google Scholar
Cruz, A. and Varandas, P.. SRB measures for partially hyperbolic attractors of endomorphisms. Ergod. Th. & Dynam. Sys. doi:10.1017/etds.2018.115.Google Scholar
Liu, P.-D., Qian, M. and Zhao, Y.. Large deviations in Axiom A endomorphisms. Proc. Roy. Soc. Edinburgh 133A(6) (2003), 1379–1388.Google Scholar
Mané, R.. Ergodic Theory and Differentiable Dynamics. Springer, Berlin, 1987.Google Scholar
Melbourne, I.. Large and moderate deviations for slowly mixing dynamical systems. Proc. Amer. Math. Soc. 137 (2009), 1735–1741.Google Scholar
Melbourne, I. and Nicol, M.. Large deviations for nonuniformly hyperbolic systems. Trans. Amer. Math. Soc. 360 (2008), 6661–6676.Google Scholar
Mihailescu, E.. Physical measures for multivalued inverse iterates near hyperbolic repellors. J. Stat. Phys. 139(5) (2010), 800–819.Google Scholar
Mihailescu, E.. Equilibrium measures, prehistories distributions and fractal dimensions for endomorphisms. Discrete Contin. Dyn. Syst. 32(7) (2012), 2485–2502.Google Scholar
Mihailescu, E. and Urbanski, M.. Entropy production for a class of inverse SRB measures. J. Stat. Phys. 150(5) (2013), 881–888.Google Scholar
Przytycki, F.. Anosov endomorphisms. Stud. Math. 3(58) (1976), 249–285.Google Scholar
Qian, M., Xie, J.-S. and Zhu, S.. Smooth Ergodic Theory for Endomorphisms. Springer, Berlin, 2009.Google Scholar
Qian, M. and Zhu, S.. SRB measures and Pesin’s entropy formula for endomorphisms. Trans. Amer. Math. Soc. 354(4) (2002), 1453–1471.Google Scholar
Rey-Bellet, l. and Young, L.-S.. Large deviations in non-uniformly hyperbolic dynamical systems. Ergod. Th. & Dynam. Sys. 28 (2008), 587–612.Google Scholar
Sternberg, S.. Lectures on Differential Geometry, 2nd edn. AMS Chelsea Publishing, Providence, RI, 1999.Google Scholar
Sumi, N.. A class of differentiable total maps which are topologically mixing. Proc. Amer. Math. Soc. 127(3) (1999), 915–924.Google Scholar
Tsujii, M.. Physical measures for partially hyperbolic surface endomorphisms. Acta Math. 194(1) (2005), 37–132.Google Scholar
Urbański, M. and Wolf, C.. SRB measures for Axiom A endomorphisms. Math. Res. Lett. 11(5–6) (2004), 785–797.Google Scholar
Varandas, P.. Non-uniform specification and large deviations for weak Gibbs measures. J. Stat. Phys. 146 (2012), 330–358.Google Scholar
Varandas, P. and Viana, M.. Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps. Ann. Inst. Henri Poincaré 27(2) (2010), 555–593.Google Scholar
Varandas, P. and Zhao, Y.. Weak specification properties and large deviations for non-additive potentials. Ergod. Th. & Dynam. Sys. 35(3) (2015), 968–993.Google Scholar
Varandas, P. and Zhao, Y.. Weak Gibbs measures: convergence to entropy, topological and geometrical aspects. Ergod. Th. & Dynam. Sys. 37(7) (2017), 2313–2336.Google Scholar
Young, L.-S.. Large deviations in dynamical systems. Trans. Amer. Math. Soc. 318(2) (1990), 525–543.Google Scholar
Young, L.-S.. What are SRB measures, and which dynamical systems have them? J. Stat. Phys. 108(5) (2002), 733–754.Google Scholar