The waiting time and dynamic partitions

Authors

Akhtam Dzhalilov, Turin Polytechnic University in TashkentFollow
Mukhriddin Khomidov, National University of UzbekistanFollow

Abstract

In the present paper we study the behaviour of normalized waiting times for linear irrational rotations. D.Kim and B.Seo investigated the waiting times for equidistance partitions. We consider waiting times with respect to dynamical partitions. The results show that limiting behaviour of waiting times essentially depend on type of partitions.

First Page

36

Last Page

51

References

1. Cornfeld I.P., Fomin S.V., Sinai Ya.G. Ergodic Theory. Springer-Verlag, Berlin, (1982).

2. Kim D.H., Seo B.K. The waiting time for irrational rotations. Nonlinearity, Vol. 16, No. 5, 1861–1868 (2003).

3. Choe G.H. Computational Ergodic theory. Algorithms and Computation in Mathematics, Vol. 13, Springer-Verlag, Berlin, Heidelberg, (2005).

4. Khinchin A.Ya. Continued Fractions. Dover Books on Mathematics, Dover Publications; Revised edition, (1997). – 112 p.

5. Wyner A.D., Ziv J. Some asymptotic proporties of the entropy of stationary ergodic data source with applications to data compression. IEEE Transactions on Information Theory, Vol. 35, Issue 6, 1250–1258 (1989).

6. Ornstein D., Weiss B. Entropy and data compression schemes. IEEE Transactions on Information Theory, Vol. 39, Issue 1, 78–83 (1993).

7. Coelho Z., de Faria E. Limit laws of entrance times for homeomorphisms of the circle. Israel J. Math., Vol. 93, Issue 1, 93–112 (1996).

8. Kim C., Kim D.H. On the law of logarithm of the recurrence time. Discrete & Continuous Dynamical Systems - A, Vol. 10, Issue 3, 581–587 (2004).

9. Barreira L., Saussol B. Hausdorff dimension of measures via Poincare recurrence. Commun. Math. Phys., Vol. 219, Issue 2, 443-463 (2001).

10. Alessandri P., Berthe V. Three distance theorems and combinatorics on words. Enseign. Math., Vol. 44, 103–32 (1998).

11. Barreire L., Saussol B. Product structure of Poincare recurrence. Ergodic Theory and Dynamical Systems, Vol. 22, Issue 1, 33–61 (2002).

12. Choe G.H. A universal law of logarithm of the recurrence time. Nonlinearity, Vol. 16, No. 3, 883–896 (2003).

13. Choe G.H., Seo B.K. Reccurence speed of multiples of an irrational number. Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 77, No. 7, 134–137 (2001).

14. Kac M. On the notion of reccurence in discrete stochastic processes. Bull. Amer. Math. Soc., Vol. 53, No. 10, 1002-1010 (1947).

15. Marton K., Shields P. Almost-sure waiting time result for weak and very weak Bernoulli processes. Ergodic Theory and Dynamical Systems, Vol. 15, Issue 5, 951–960 (1995).

16. Rockett A., Szusz P. Continued Fractions. World Scientific Publishing Company, Singapore, (1992). – 198 p.

17. Saussol B., Troubetzkoy S., Vaienti S. Reccurence, dimensions and Lyapunov exponents. Journal of Statistical Physics, Vol. 106, Issue 3–4, 623–634 (2002).

18. Slater N.B. Gaps and steps for the sequnce n θ mod 1. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 63, Issue 4, 1115–1123 (1967).

19. Shields P. Waiting times: positive and negative result on the Wyner-Ziv problem. Journal of Theoretical Probability, Vol. 6, Issue 3, 499–519 (1993).

20. de Melo W., van Strein S. One Dimensional dynamics. Springer-Verlag, Berlin, (1993).

Recommended Citation

Dzhalilov, Akhtam and Khomidov, Mukhriddin (2019) "The waiting time and dynamic partitions," Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences: Vol. 2: Iss. 1, Article 3.
DOI: https://doi.org/10.56017/2181-1318.1018