phone +7 (3412) 91 60 92
mail imi@udsu.ru
ru-RU Russian
Home » Issues » Archive of Issues » 2023 - 2 issue » pp. 197-211

Archive of Issues

Uzbekistan Tashkent
Year
2023
Volume
33
Issue
2
Pages
197-211
>>
Section Mathematics
Title Hitting functions for mixed partitions
Author(-s) Dzhalilov A.A.a, Khomidov M.K.b
Affiliations Turin Polytechnic University in Tashkenta, National University of Uzbekistanb
Abstract Let $T_{\rho}$ be an irrational rotation on a unit circle $S^{1}\simeq [0,1)$. Consider the sequence $\{\mathcal{P}_{n}\}$ of increasing partitions on $S^{1}$. Define the hitting times $N_{n}(\mathcal{P}_n;x,y):= \inf\{j\geq 1\mid T^{j}_{\rho}(y)\in P_{n}(x)\}$, where $P_{n}(x)$ is an element of $\mathcal{P}_{n}$ containing $x$. D. Kim and B. Seo in [9] proved that the rescaled hitting times $K_n(\mathcal{Q}_n;x,y):= \frac{\log N_n(\mathcal{Q}_n;x,y)}{n}$ a.e. (with respect to the Lebesgue measure) converge to $\log2$, where the sequence of partitions $\{\mathcal{Q}_n\}$ is associated with chaotic map $f_{2}(x):=2x \bmod 1$. The map $f_{2}(x)$ has positive entropy $\log2$. A natural question is what if the sequence of partitions $\{\mathcal{P}_n\}$ is associated with a map with zero entropy. In present work we study the behavior of $K_n(\tau_n;x,y)$ with the sequence of mixed partitions $\{\tau_{n}\}$ such that $ \mathcal{P}_{n}\cap [0,\frac{1}{2}]$ is associated with map $f_{2}$ and $\mathcal{D}_{n}\cap [\frac{1}{2},1]$ is associated with irrational rotation $T_{\rho}$. It is proved that $K_n(\tau_n;x,y)$ a.e. converges to a piecewise constant function with two values. Also, it is shown that there are some irrational rotations that exhibit different behavior.
Keywords irrational rotation, hitting time, dynamical partition, limit theorem
UDC 517.938
MSC 37C05, 37C15, 37E05, 37E10, 37E20, 37B10
DOI 10.35634/vm230201
Received 3 October 2022
Language English
Citation Dzhalilov A.A., Khomidov M.K. Hitting functions for mixed partitions, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2023, vol. 33, issue 2, pp. 197-211.
References
  1. Hussain M., Li Bing, Simmons D., Wang Baowei. Dynamical Borel-Cantelli lemma for recurrence theory, arXiv:2009.03515 [math.DS], 2021. https://doi.org/10.48550/arXiv.2009.03515
  2. Kleinbock D., Yu Shucheng. A dynamical Borel-Cantelli lemma via improvements to Dirichlet's theorem, Moscow Journal of Combinatorics and Number Theory, 2020, vol. 9, no. 2, pp. 101-122. https://doi.org/10.2140/moscow.2020.9.101
  3. Maucourant F. Dynamical Borel-Cantelli lemma for hyperbolic spaces, Israel Journal of Mathematics, 2006, vol. 152, issue 1, pp. 143-155. https://doi.org/10.1007/BF02771980
  4. Chernov N., Kleinbock D. Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel Journal of Mathematics, 2001, vol. 122, issue 1, pp. 1-27. https://doi.org/10.1007/BF02809888
  5. Athreya J.S. Logarithm laws and shrinking target properties, Proceedings - Mathematical Sciences, 2009, vol. 119, issue 4, pp. 541-557. https://doi.org/10.1007/s12044-009-0044-x
  6. Li Bing, Wang Bao-Wei, Wu Jun, Xu Jian. The shrinking target problem in the dynamical system of continued fractions, Proceedings of the London Mathematical Society, 2014, vol. 108, issue 1, pp. 159-186. https://doi.org/10.1112/plms/pdt017
  7. Fayad B. Mixing in the absence of the shrinking target property, Bulletin of the London Mathematical Society, 2006, vol. 38, issue 5, pp. 829-838. https://doi.org/10.1112/S0024609306018546
  8. Galatolo S., Kim Dong Han. The dynamical Borel-Cantelli lemma and the waiting time problems, Indagationes Mathematicae, 2007, vol. 18, issue 3, pp. 421-434. https://doi.org/10.1016/S0019-3577(07)80031-0
  9. Kim Dong Han, Seo Byoung Ki. The waiting time for irrational rotations, Nonlinearity, 2003, vol. 16, issue 5, pp. 1861-1868. https://doi.org/10.1088/0951-7715/16/5/318
  10. Choe Geon Ho. Computational ergodic theory, Berlin-Heidelberg: Springer, 2005. https://doi.org/10.1007/b138894
  11. Barreira L., Saussol B. Hausdorff dimension of measures via Poincaré recurrence, Communications in Mathematical Physics, 2001, vol. 219, issue 2, pp. 443-463. https://doi.org/10.1007/s002200100427
  12. Wyner A.D., Ziv J. Some asymptotic properties of the entropy of a stationary ergodic data source with applications to data compression, IEEE Transactions on Information Theory, 1989, vol. 35, issue 6, pp. 1250-1258. https://doi.org/10.1109/18.45281
  13. Ornstein D.S., Weiss B. Entropy and data compression schemes, IEEE Transactions on Information Theory, 1993, vol. 39, issue 1, pp. 78-83. https://doi.org/10.1109/18.179344
  14. Khomidov M.K. A note on behaviour of the first return times for irrational rotations, Uzbek Mathematical Journal, 2021, vol. 65, issue 4, pp. 79-88. https://zbmath.org/1499.37080
  15. Khanin K.M., Sinai Ya.G. A new proof of M. Herman's theorem, Communications in Mathematical Physics, 1987, vol. 112, issue 1, pp. 89-101. https://doi.org/10.1007/BF01217681
  16. Kim Chihurn, Kim Dong Han. On the law of logarithm of the recurrence time, Discrete and Continuous Dynamical Systems - A, 2004, vol. 10, issue 3, pp. 581-587. https://doi.org/10.3934/dcds.2004.10.581
  17. Saussol B., Troubetzkoy S., Vaienti S. Reccurence, dimensions and Lyapunov exponents, Journal of Statistical Physics, 2002, vol. 106, issue 3, pp. 623-634. https://doi.org/10.1023/A:1013710422755
  18. Kac M. On the notion of recurrence in discrete stochastic processes, Bulletin of the American Mathematical Society, 1947, vol. 53, issue 10, pp. 1002-1010. https://doi.org/10.1090/S0002-9904-1947-08927-8
  19. Slater N.B. Gaps and steps for the sequnce $n\theta \mod 1$, Mathematical Proceedings of the Cambridge Philosophical Society, 1967, vol. 63, issue 4, pp. 1115-1123. https://doi.org/10.1017/S0305004100042195
  20. Coelho Z., de Faria E. Limit laws of entrance times for homeomorphisms of the circle, Israel Journal of Mathematics, 1996, vol. 93, issue 1, pp. 93-112. https://doi.org/10.1007/BF02761095
  21. Carletti T., Galatolo S. Numerical estimates of local dimension by waiting time and quantitative recurrence, Physica A: Statistical Mechanics and its Applications, 2006, vol. 364, pp. 120-128. https://doi.org/10.1016/j.physa.2005.10.003
  22. Kim Dong Han. The dynamical Borel-Cantelli lemma for interval maps, Discrete and Continuous Dynamical Systems - A, 2007, vol. 17, issue 4, pp. 891-900. https://doi.org/10.3934/dcds.2007.17.891
  23. Kornfel'd I.P., Sinai Ya.G., Fomin S.V. Ergodicheskaya teoriya (Ergodic theory), Moscow: Nauka, 1980.
Full text
/doc/issues-2023/23-02-01.pdf
Next article >>