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Polynomial decay of correlations for almost Anosov diffeomorphisms

Published online by Cambridge University Press:  17 August 2017

XU ZHANG  and
HUYI HU
XU ZHANG
Affiliation:
Department of Mathematics, Shandong University, Weihai, Shandong 264209, China email xuzhang08@gmail.com
HUYI HU
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA email hu@math.msu.edu
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Abstract

We investigate the polynomial lower and upper bounds for decay of correlations of a class of two-dimensional almost Anosov diffeomorphisms with respect to their Sinai–Ruelle–Bowen (SRB) measures. The degrees of the bounds are determined by the expansion and contraction rates as the orbits approach the indifferent fixed point, and are expressed by using coefficients of the third-order terms in the Taylor expansions of the diffeomorphisms at the indifferent fixed point.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 3 , March 2019 , pp. 832 - 864
Copyright
© Cambridge University Press, 2017 

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References

Aaronson, J., Denker, M. and Urbanski, M.. Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. Amer. Math. Soc. 337 (1993), 495–548.Google Scholar
Alves, J. and Azevedo, D.. Statistical properties of diffeomorphisms with weak invariant manifolds. Discrete Contin. Dyn. Syst. 36 (2016), 1–41.Google Scholar
Bruin, H., Holland, M. and Melbourne, I.. Subexponential decay of correlations for compact group extensions of non-uniformly expanding systems. Ergod. Th. & Dynam. Sys. 25 (2005), 1719–1738.Google Scholar
Chart, S. W.. Limit theorems for generalized Baker’s transformations. Preprint, 2016, arXiv:1606.02004.Google Scholar
Chernov, N., Vaienti, S. and Zhang, H.. Lower bound for decay rates of correlations of nonuniformly hyperbolic systems. Preprint.Google Scholar
Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, NJ, 1981.Google Scholar
Gouëzel, S.. Sharp polynomial estimates for the decay of correlations. Israel J. Math. 139 (2004), 29–65.Google Scholar
Hu, H.. Conditions for the existence of SBR measures for ‘almost Anosov’ diffeomorphisms. Trans. Amer. Math. Soc. 352(5) (1999), 2331–2367.Google Scholar
Hu, H.. Decay of correlations for piecewise smooth maps with indifferent fixed points. Ergod. Th. & Dynam. Sys. 24 (2004), 495–524.Google Scholar
Hu, H. and Vaienti, S.. Absolutely continuous invariant measures for some non-uniformly expanding maps. Ergod. Th. & Dynam. Sys. 29 (2009), 1185–1215.Google Scholar
Hu, H. and Vaienti, S.. Polynomial bounds for the decay of correlations in non-uniformly expanding maps. Preprint, 2013, arXiv:1307.0359.Google Scholar
Hu, H. and Young, L. S.. Nonexistence of SBR measures for some diffeomorphisms that are ‘almost Anosov’. Ergod. Th. & Dynam. Sys. 15 (1995), 67–76.Google Scholar
Ledrappier, F. and Strelcyn, J. M.. Estimation from below in Pesin’s entropy formula. Ergod. Th. & Dynam. Sys. 2 (1982), 203–219.Google Scholar
Liverani, C. and Martens, M.. Convergence to equilibrium for intermittent symplectic maps. Comm. Math. Phys. 260 (2005), 527–556.Google Scholar
Liverani, C., Saussol, B. and Vaienti, S.. A probabilistic approach to intermittency. Ergod. Th. & Dynam. Sys. 19 (1999), 671–685.Google Scholar
Liverani, C. and Terhesiu, D.. Mixing for some non-uniformly hyperbolic systems. Ann. Henri Poincaré 17 (2016), 179–226.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
Melbourne, I. and Terhesiu, D.. Decay of correlation for nonuniformly expanding systems with general return times. Ergod. Th. & Dynam. Sys. 34 (2014), 893–918.Google Scholar
Peligrad, M., Utev, S. and Wu, W. B.. A maximal L p inequality for stationary sequences and its applications. Proc. Amer. Math. Soc. 135 (2007), 541–550.Google Scholar
Pesin, Ya. B.. Families of invariant manifolds corresponding to non-zero characteristic exponents. Math. USSR-Izv. 10 (1978), 1261–1305.Google Scholar
Pollicott, M. and Sharp, R.. Large deviations for intermittent maps. Nonlinearity 22 (2009), 2079–2092.Google Scholar
Pollicott, M. and Yuri, M.. Statistical properties of maps with indifferent periodic points. Comm. Math. Phys. 217 (2001), 503–520.Google Scholar
Rohlin, V. A.. Lectures on the theory of entropy of transformations with invariant measures. Russian Math. Surveys 22 (1967), 1–54.Google Scholar
Saks, S.. Theory of the Integral. Polish Mathematical Society, Warszawa-Lwów, 1937.Google Scholar
Sarig, O.. Subexponential decay of correlations. Invent. Math. 150 (2002), 629–653.Google Scholar
Sarig, O.. Introduction to the transfer operator method. Second Brazilian School on Dynamical Systems (Lecture Notes) . 2012. Available at http://www.weizmann.ac.il/math/sarigo/sites/math.sarigo/files/uploads/transferoperatorcourse_0.pdf.Google Scholar
Young, L. S.. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. 147 (1998), 585–650.Google Scholar
Young, L. S.. Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153–188.Google Scholar
Zhang, H.. Mixing rate for rectangular billiards with diamond obstacles. Amer. J. Math. Management Sci. 30 (2010), 53–65.Google Scholar