The first return time properties of an irrational rotation

Kim, Dong Han; Park, Kyewon

doi:10.1090/S0002-9939-08-09388-X

The first return time properties of an irrational rotation
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by Dong Han Kim and Kyewon Koh Park

Proc. Amer. Math. Soc. 136 (2008), 3941-3951

DOI: https://doi.org/10.1090/S0002-9939-08-09388-X

Published electronically: June 2, 2008

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Abstract:

If an ergodic system has positive entropy, then the Shannon-McMillan-Breiman theorem provides a relationship between the entropy and the size of an atom of the iterated partition. The system also has Ornstein-Weiss’ first return time property, which offers a method of computing the entropy via an orbit. We consider irrational rotations which are the simplest model of zero entropy. We prove that almost every irrational rotation has the analogous properties if properly normalized. However there are some irrational rotations that exhibit different behavior.

References

Boris Adamczewski and Julien Cassaigne, Diophantine properties of real numbers generated by finite automata, Compos. Math. 142 (2006), no. 6, 1351–1372. MR 2278750, DOI 10.1112/S0010437X06002247
Pascal Alessandri and Valérie Berthé, Three distance theorems and combinatorics on words, Enseign. Math. (2) 44 (1998), no. 1-2, 103–132. MR 1643286
L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincaré recurrence, Comm. Math. Phys. 219 (2001), no. 2, 443–463. MR 1833809, DOI 10.1007/s002200100427
L. Barreira and B. Saussol, Product structure of Poincaré recurrence, Ergodic Theory Dynam. Systems 22 (2002), no. 1, 33–61. MR 1889564, DOI 10.1017/S0143385702000020
Mike Boyle and Douglas Lind, Expansive subdynamics, Trans. Amer. Math. Soc. 349 (1997), no. 1, 55–102. MR 1355295, DOI 10.1090/S0002-9947-97-01634-6
Srečko Brlek, Enumeration of factors in the Thue-Morse word, Discrete Appl. Math. 24 (1989), no. 1-3, 83–96. First Montreal Conference on Combinatorics and Computer Science, 1987. MR 1011264, DOI 10.1016/0166-218X(92)90274-E
Geon Ho Choe and Byoung Ki Seo, Recurrence speed of multiples of an irrational number, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), no. 7, 134–137. MR 1857291
Alan Cobham, Uniform tag sequences, Math. Systems Theory 6 (1972), 164–192. MR 457011, DOI 10.1007/BF01706087
Samuel Eilenberg, Automata, languages, and machines. Vol. A, Pure and Applied Mathematics, Vol. 58, Academic Press [Harcourt Brace Jovanovich, Publishers], New York, 1974. MR 0530382
Sébastien Ferenczi and Kyewon Koh Park, Entropy dimensions and a class of constructive examples, Discrete Contin. Dyn. Syst. 17 (2007), no. 1, 133–141. MR 2257422, DOI 10.3934/dcds.2007.17.133
M. Kac, On the notion of recurrence in discrete stochastic processes, Bull. Amer. Math. Soc. 53 (1947), 1002–1010. MR 22323, DOI 10.1090/S0002-9904-1947-08927-8
Anatole Katok and Jean-Paul Thouvenot, Slow entropy type invariants and smooth realization of commuting measure-preserving transformations, Ann. Inst. H. Poincaré Probab. Statist. 33 (1997), no. 3, 323–338 (English, with English and French summaries). MR 1457054, DOI 10.1016/S0246-0203(97)80094-5
A. Ya. Khinchin, Continued fractions, University of Chicago Press, Chicago, Ill.-London, 1964. MR 0161833
Chihurn Kim and Dong Han Kim, On the law of logarithm of the recurrence time, Discrete Contin. Dyn. Syst. 10 (2004), no. 3, 581–587. MR 2018868, DOI 10.3934/dcds.2004.10.581
Dong Han Kim, The recurrence time for irrational rotations, Osaka J. Math. 43 (2006), no. 2, 351–364. MR 2262339
Dong Han Kim and Byoung Ki Seo, The waiting time for irrational rotations, Nonlinearity 16 (2003), no. 5, 1861–1868. MR 1999584, DOI 10.1088/0951-7715/16/5/318
M. Lothaire, Algebraic combinatorics on words, Encyclopedia of Mathematics and its Applications, vol. 90, Cambridge University Press, Cambridge, 2002. A collective work by Jean Berstel, Dominique Perrin, Patrice Seebold, Julien Cassaigne, Aldo De Luca, Steffano Varricchio, Alain Lascoux, Bernard Leclerc, Jean-Yves Thibon, Veronique Bruyere, Christiane Frougny, Filippo Mignosi, Antonio Restivo, Christophe Reutenauer, Dominique Foata, Guo-Niu Han, Jacques Desarmenien, Volker Diekert, Tero Harju, Juhani Karhumaki and Wojciech Plandowski; With a preface by Berstel and Perrin. MR 1905123, DOI 10.1017/CBO9781107326019
John Milnor, On the entropy geometry of cellular automata, Complex Systems 2 (1988), no. 3, 357–385. MR 955558
Donald Samuel Ornstein and Benjamin Weiss, Entropy and data compression schemes, IEEE Trans. Inform. Theory 39 (1993), no. 1, 78–83. MR 1211492, DOI 10.1109/18.179344
Kyewon Koh Park, On directional entropy functions, Israel J. Math. 113 (1999), 243–267. MR 1729449, DOI 10.1007/BF02780179
Andrew M. Rockett and Peter Szüsz, Continued fractions, World Scientific Publishing Co., Inc., River Edge, NJ, 1992. MR 1188878, DOI 10.1142/1725
B. Saussol, S. Troubetzkoy, and S. Vaienti, Recurrence, dimensions, and Lyapunov exponents, J. Statist. Phys. 106 (2002), no. 3-4, 623–634. MR 1884547, DOI 10.1023/A:1013710422755
Noel B. Slater, Gaps and steps for the sequence $n\theta \ \textrm {mod}\ 1$, Proc. Cambridge Philos. Soc. 63 (1967), 1115–1123. MR 217019, DOI 10.1017/s0305004100042195
Vladimir G. Sprindžuk, Metric theory of Diophantine approximations, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C.; John Wiley & Sons, New York-Toronto-London, 1979. Translated from the Russian and edited by Richard A. Silverman; With a foreword by Donald J. Newman. MR 548467
Aaron D. Wyner and Jacob Ziv, Some asymptotic properties of the entropy of a stationary ergodic data source with applications to data compression, IEEE Trans. Inform. Theory 35 (1989), no. 6, 1250–1258. MR 1036629, DOI 10.1109/18.45281