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Limit Theorems for Self-Similar Tilings

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Abstract

We study deviation of ergodic averages for dynamical systems given by self-similar tilings on the plane and in higher dimensions. The main object of our paper is a special family of finitely-additive measures for our systems. An asymptotic formula is given for ergodic integrals in terms of these finitely-additive measures, and, as a corollary, limit theorems are obtained for dynamical systems given by self-similar tilings.

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Authors and Affiliations

  1. Laboratoire d’Analyse, Topologie, Probabilités, Aix-Marseille Université, CNRS, Marseille, France

    Alexander I. Bufetov

  2. Steklov Institute, Moscow, Russia

    Alexander I. Bufetov

  3. The Institute for Information Transmission Problems, Moscow, Russia

    Alexander I. Bufetov

  4. National Research University Higher School of Economics, Moscow, Russia

    Alexander I. Bufetov

  5. Department of Mathematics, Rice University, Houston, TX, 77005, USA

    Alexander I. Bufetov

  6. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195, USA

    Boris Solomyak

Authors
  1. Alexander I. Bufetov
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  2. Boris Solomyak
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Correspondence to Alexander I. Bufetov.

Additional information

Communicated by G. Gallavotti

A. B. is an Alfred P. Sloan Fellow, a Dynasty Foundation Fellow, and an IUM-Simons Fellow. He is supported in part by the Grant MK 6734.2012.1 of the President of the Russian Federation, by the Programme on Dynamical Systems and Mathematical Control Theory of the Presidium of the Russian Academy of Sciences, by RFBR-CNRS grant 10-01-93115 and by the RFBR grant 11-01-00654.

B. S. is supported in part by NSF grant DMS-0968879.

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Bufetov, A.I., Solomyak, B. Limit Theorems for Self-Similar Tilings. Commun. Math. Phys. 319, 761–789 (2013). https://doi.org/10.1007/s00220-012-1624-7

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