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On mixing and the local central limit theorem for hyperbolic flows

Published online by Cambridge University Press:  11 May 2018

DMITRY DOLGOPYAT  and
PÉTER NÁNDORI
DMITRY DOLGOPYAT
Affiliation:
University of Maryland, Department of Mathematics, 4176 Campus Drive College Park, MD 20742-4015, USA email [email protected], [email protected]
PÉTER NÁNDORI
Affiliation:
University of Maryland, Department of Mathematics, 4176 Campus Drive College Park, MD 20742-4015, USA email [email protected], [email protected]
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Abstract

We formulate abstract conditions under which a suspension flow satisfies the local central limit theorem. We check the validity of these conditions for several systems including reward renewal processes, Axiom A flows, as well as the systems admitting Young’s tower, such as Sinai’s billiard with finite horizon, suspensions over Pomeau–Manneville maps, and geometric Lorenz attractors.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 1 , January 2020 , pp. 142 - 174
Copyright
© Cambridge University Press, 2018 

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References

Aaronson, J. and Denker, M.. Local limit theorems for partial sums of stationary sequences generated by Gibbs–Markov maps. Stoch. Dyn. 1 (2001), 193–237.10.1142/S0219493701000114Google Scholar
Aaronson, J. and Nakada, H.. On multiple recurrence and other properties of ‘nice’ infinite measure-preserving transformations. Ergod. Th. & Dynam. Sys. 37 (2017), 1345–1368.Google Scholar
Bowen, R.. Symbolic dynamics for hyperbolic flows. Amer. J. Math. 95 (1973), 429–460.Google Scholar
Bowen, R. and Ruelle, D.. The ergodic theory of axiom A flows. Invent. Math. 29 (1975), 181–202.10.1007/BF01389848Google Scholar
Chernov, N. I. and Dolgopyat, D. I.. Anomalous current in periodic Lorentz gases with an infinite horizon. Russian Math. Surveys 64 (2009), 651–699.Google Scholar
Dolgopyat, D. and Nándori, P.. Non equilibrium density profiles in Lorentz tubes with thermostated boundaries. Commun. Pure Appl. Math. 69 (2016), 649–692.Google Scholar
Dolgopyat, D. and Nándori, P.. Infinite measure renewal theorem and related results. Preprint, 2017.10.1112/blms.12217Google Scholar
Denker, M. and Philipp, W.. Approximation by Brownian motion for Gibbs measures and flows under a function. Ergod. Th. & Dynam. Sys. 4 (1984), 541–552.Google Scholar
Feller, W.. An Introduction to Probability Theory and its Applications, Vol 2. Wiley, New York, NY, 1968.Google Scholar
Gouëzel, S.. Berry–Esseen theorem and local limit theorem for non uniformly expanding maps. Ann. Inst. H. Poincaré 41 (2005), 997–1024.Google Scholar
Guivarc’h, Y.. Propriétés ergodiques, en mesure infinie, de certains systèmes dynamiques fibrés. Ergod. Th. & Dynam. Sys. 9 (1989), 433–453.Google Scholar
Guivarc’h, Y. and Hardy, J.. Theoremes limites pour une classe de chaines de Markov et applications aux diffeomorphismes d’Anosov. Ann. Inst. H. Poincaré 24 (1988), 73–98.Google Scholar
Glynn, P. W. and Whitt, W.. Limit theorems for cumulative processes. Stochastic Process. Appl. 47 (1993), 299–314.10.1016/0304-4149(93)90019-ZGoogle Scholar
Holland, M. and Melbourne, I.. Central limit theorems and invariance principles for Lorenz attractors. J. Lond. Math. Soc. 76 (2007), 345–364.10.1112/jlms/jdm060Google Scholar
Iwata, Y.. A generalized local limit theorem for mixing semi-flows. Hokkaido Math. J. 37 (2008), 215–240.10.14492/hokmj/1253539585Google Scholar
Ionescu-Tulcea, C. T. and Marinescu, G.. Theorie ergodique pour des classes d’operations non completement continues. Ann. of Math. (2) 47 (1950), 140–147.Google Scholar
Luzzato, S., Melbourne, I. and Paccaut, F.. The Lorenz attractor is mixing. Comm. Math. Phys. 260 (2005), 393–401.Google Scholar
Liverani, C., Saussol, B. and Vaienti, S.. A probabilistic approach to intermittency. Ergod. Th. & Dynam. Sys. 19 (1999), 671–685.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 Török, A.. Statistical limit theorems for suspension flows. Israel J. Math. 144 (2004), 191–209.Google Scholar
Petrov, V. V.. Limit theorems in probability theory: Sequences of independent random variables. Oxford Studies in Probability. Vol. 4. Oxford Science Publications, Clarendon Press, Oxford, 1995.Google Scholar
Pène, F.. Planar Lorentz process in random scenery. Ann. Inst. H. Poincaré 45 (2009), 818–839.Google Scholar
Pesin, Y., Senti, S. and Zhang, K.. Thermodynamics of the Katok map. Ergod. Th. & Dynam. Sys. doi:10.1017/etds.2017.35. Published online 28 June 2017.Google Scholar
Rvaceva, E. L.. On the domains of attraction of multidimensional distributions. Sel. Trans. Math. Statist. Probab. Theory 2 (1962), 183–205.Google Scholar
Ratner, M.. The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature. Israel J. Math. 16 (1973), 181–197.10.1007/BF02757869Google Scholar
Rousseau-Egele, J.. Un theoreme de la limite local pour une classe de transformations dilatantes et monotones par morceaux. Ann. Probab. 11 (1983), 772–788.Google Scholar
Stone, C. J.. A local limit theorem for non-lattice multi-dimensional distribution functions. Ann. Math. Statist. 36 (1965), 546–551.Google Scholar
Szász, D. and Varjú, T.. Local limit theorem for the Lorentz process and its recurrence in the plane. Ergod. Th. & Dynam. Sys. 24 (2004), 254–278.Google Scholar
Szász, D. and Varjú, T.. Limit laws and recurrence for the planar Lorentz process with infinite horizon. J. Stat. Phys. 129 (2007), 59–80.10.1007/s10955-007-9367-0Google Scholar
Wolfe, S. J.. On the local behavior of characteristic functions. Ann. Probab. 1 (1973), 862–866.Google Scholar
Waddington, S.. Large deviation asymptotics for Anosov flows. Ann. Inst. H. Poincaré 13 (1996), 445–484.10.1016/S0294-1449(16)30110-XGoogle Scholar
Young, L-S.. Statistical properties of systems with some hyperbolicity including certain billiards. Ann. of Math. (2) 147 (1998), 585–650.Google Scholar
Young, L-S.. Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153–188.Google Scholar