Local Limit Theorem for the Lorentz Process and Its Recurrence in the Plane
Tamás Varjú,Domokos Szász +1 more
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For planar Lorentz processes with a finite horizon, Schmidt and Conze as discussed by the authors proved a local central limit theorem (CLT) and recurrence for the case d = 2, finite horizon.Abstract:
For Young systems, i.e. for hyperbolic systems without/with singularities satisfying Young's axioms (Lai-Sang Young, Ann. Math.147 (1998), 585–650), which imply exponential decay of correlations and the central limit theorem (CLT), a local CLT is proven. In fact, a unified version of the local CLT is found, covering, among others, the absolutely continuous and arithmetic cases. For planar Lorentz process with a finite horizon, this result implies (a) a local CLT and (b) recurrence. For the latter case (d = 2, finite horizon), combining the global CLT with abstract ergodic theoretic ideas, K. Schmidt (C. R. Acad. Sci. Paris Ser. 1 Math. 372(9) (1998), 837–842) and J.-P. Conze (Ergod. Th. & Dynam. Sys.19(5) (1999), 1233–1245) could already establish recurrence.read more
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Large deviations in non-uniformly hyperbolic dynamical systems
Luc Rey-Bellet,Lai Sang Young +1 more
TL;DR: In this article, large deviation principles for ergodic averages of dynamical systems admitting Markov tower extensions with exponential return times were proved for piecewise hyperbolic diffeomorphisms, dispersing billiards including Lorentz gases, and strange attractors of rank one including Hattractors.
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Berry-Esseen theorem and local limit theorem for non uniformly expanding maps
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Limit Laws and Recurrence for the Planar Lorentz Process with Infinite Horizon
Domokos Szász,Tamás Varjú +1 more
TL;DR: Szasz et al. as discussed by the authors showed that the free flight vector of the planar, infinite horizon, periodic Lorentz process belongs to the non-standard domain of attraction of the Gaussian law.
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Limit Theorems in the Stadium Billiard
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TL;DR: In this article, it was shown that the Birkhoff sums for almost every relevant observable in the stadium billiard obey a non-standard limit law, i.e., the usual central limit theorem holds for an observable if and only if its integral along a one-codimensional invariant set vanishes, otherwise a Open image in new window normalization is needed.
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Recurrence properties of planar Lorentz process
TL;DR: In this article, the first return and first hitting times, local times, and first intersection times for planar finite-horizon Lorentz processes with a periodic configuration of scatterers were studied.
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