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Limit Theorems for Dispersing Billiards with Cusps

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Abstract

Dispersing billiards with cusps are deterministic dynamical systems with a mild degree of chaos, exhibiting “intermittent” behavior that alternates between regular and chaotic patterns. Their statistical properties are therefore weak and delicate. They are characterized by a slow (power-law) decay of correlations, and as a result the classical central limit theorem fails. We prove that a non-classical central limit theorem holds, with a scaling factor of \({\sqrt{n\log n}}\) replacing the standard \({\sqrt{n}}\) . We also derive the respective Weak Invariance Principle, and we identify the class of observables for which the classical CLT still holds.

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Authors and Affiliations

  1. Institute of Mathematics, Budapest University of Technology and Economics, H-1111, Egry Jozsef u. 1, Budapest, Hungary

    P. Bálint

  2. Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL, 35294, USA

    N. Chernov

  3. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    D. Dolgopyat

Authors
  1. P. Bálint
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  2. N. Chernov
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  3. D. Dolgopyat
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Correspondence to N. Chernov.

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Communicated by G. Gallavotti

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Bálint, P., Chernov, N. & Dolgopyat, D. Limit Theorems for Dispersing Billiards with Cusps. Commun. Math. Phys. 308, 479–510 (2011). https://doi.org/10.1007/s00220-011-1342-6

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  • DOI: https://doi.org/10.1007/s00220-011-1342-6

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