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An optimal transportation approach to the decay of correlations for non-uniformly expanding maps

Published online by Cambridge University Press:  13 August 2018

BENOÎT R. KLOECKNER Open the ORCID record for BENOÎT R. KLOECKNER [Opens in a new window]
BENOÎT R. KLOECKNER*
Affiliation:
Université Paris-Est, Laboratoire d’Analyse et de Matématiques Appliquées (UMR 8050), UPEM, UPEC, CNRS, F-94010, Créteil, France email benoit.kloeckner@u-pec.fr
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Abstract

We consider the transfer operators of non-uniformly expanding maps for potentials of various regularity, and show that a specific property of potentials (‘flatness’) implies a Ruelle–Perron–Frobenius theorem and a decay of the transfer operator of the same speed as that entailed by the constant potential. The method relies neither on Markov partitions nor on inducing, but on functional analysis and duality, through the simplest principles of optimal transportation. As an application, we notably show that for any map of the circle which is expanding outside an arbitrarily flat neutral point, the set of Hölder potentials exhibiting a spectral gap is dense in the uniform topology. The method applies in a variety of situations, including Pomeau–Manneville maps with regular enough potentials, or uniformly expanding maps of low regularity with their natural potential; we also recover in a united fashion variants of several previous results.

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 3 , March 2020 , pp. 714 - 750
Copyright
© Cambridge University Press, 2018 

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References

Bruin, H. and Todd, M.. Equilibrium states for interval maps: potentials with sup𝜙-inf𝜙 < h top(f). Comm. Math. Phys. 283(3) (2008), 579–611.Google Scholar
Cyr, V. and Sarig, O.. Spectral gap and transience for Ruelle operators on countable Markov shifts. Comm. Math. Phys. 292(3) (2009), 637–666.Google Scholar
Castro, A. and Varandas, P.. Equilibrium states for non-uniformly expanding maps: decay of correlations and strong stability. Ann. Inst. H. Poincaré Anal. Non Linéaire 30(2) (2013), 225–249.Google Scholar
Denker, M. and Urbański, M.. On the existence of conformal measures. Trans. Amer. Math. Soc. 328(2) (1991), 563–587.Google Scholar
Fan, A. and Jiang, Y.. On Ruelle-Perron-Frobenius operators. I. Ruelle theorem. Comm. Math. Phys. 223(1) (2001), 125–141.Google Scholar
Fan, A. and Jiang, Y.. On Ruelle-Perron-Frobenius operators. II. Convergence speeds. Comm. Math. Phys. 223(1) (2001), 143–159.Google Scholar
Giulietti, P., Kloeckner, B. R., Lopes, A. O. and Marcon, D.. The calculus of thermodynamical formalism. J. Eur. Math. Soc. (JEMS) doi:10.4171/JEMS/814, Published online 9 July 2018.Google Scholar
Gouëzel, S.. Sharp polynomial estimates for the decay of correlations. Israel J. Math. 139(1) (2004), 29–65.Google Scholar
Gouëzel, S.. Berry-Esseen theorem and local limit theorem for non uniformly expanding maps. Ann. Inst. Henri Poincaré Probab. Stat. 41(6) (2005), 997–1024.Google Scholar
Guihéneuf, P.-A.. Dynamical properties of spatial discretizations of a generic homeomorphism. Ergod. Th. & Dynam. Sys. 35(5) (2015), 1474–1523.Google Scholar
Hofbauer, F. and Keller, G.. Equilibrium states for piecewise monotonic transformations. Ergod. Th. & Dynam. Sys. 2(1) (1982), 23–43.Google Scholar
Holland, M.. Slowly mixing systems and intermittency maps. Ergod. Th. & Dynam. Sys. 25(1) (2005), 133–159.Google Scholar
Hu, H.. Decay of correlations for piecewise smooth maps with indifferent fixed points. Ergod. Th. & Dynam. Sys. 24(2) (2004), 495–524.Google Scholar
Kloeckner, B. R.. Effective high-temperature estimates for intermittent maps. doi:10.1017/ etds.2017.111. Published online 12 December 2017.Google Scholar
Kloeckner, B. R., Lopes, A. O. and Stadlbauer, M.. Contraction in the Wasserstein metric for some Markov chains, and applications to the dynamics of expanding maps. Nonlinearity 28(11) (2015), 4117–4137.Google Scholar
Li, H. and Rivera-Letelier, J.. Equilibrium states of interval maps for hyperbolic potentials. Nonlinearity 27(8) (2014), 1779–1804.Google Scholar
Li, H. and Rivera-Letelier, J.. Equilibrium states of weakly hyperbolic one-dimensional maps for Hölder potentials. Comm. Math. Phys. 328(1) (2014), 397–419.Google Scholar
Liverani, C., Saussol, B. and Vaienti, S.. Conformal measure and decay of correlation for covering weighted systems. Ergod. Th. & Dynam. Sys. 18(6) (1998), 1399–1420.Google Scholar
Liverani, C., Saussol, B. and Vaienti, S.. A probabilistic approach to intermittency. Ergod. Th. & Dynam. Sys. 19(3) (1999), 671–685.Google Scholar
Lynch, V.. Decay of correlations for non-Hölder observables. Discrete Contin. Dyn. Syst. 16(1) (2006), 19–46.Google Scholar
Melbourne, I. and Nicol, M.. Almost sure invariance principle for nonuniformly hyperbolic systems. Comm. Math. Phys. 260(1) (2005), 131–146.Google Scholar
Pomeau, Y. and Manneville, P.. Intermittent transition to turbulence in dissipative dynamical systems. Comm. Math. Phys. 74(2) (1980), 189–197.Google Scholar
Prellberg, T. and Slawny, J.. Maps of intervals with indifferent fixed points: thermodynamic formalism and phase transitions. J. Stat. Phys. 66(1–2) (1992), 503–514.Google Scholar
Sarig, O.. Subexponential decay of correlations. Invent. Math. 150(3) (2002), 629–653.Google Scholar
Tyran-Kamińska, M.. An invariance principle for maps with polynomial decay of correlations. Comm. Math. Phys. 260(1) (2005), 1–15.Google Scholar
Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79) . Springer, New York, 1982.Google Scholar
Young, L.-S.. Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. (2) 147(3) (1998), 585–650.Google Scholar
Young, L.-S.. Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153–188.Google Scholar