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MATHEMATICS
Hitting functions for mixed partitions
A. A. Dzhalilova, M. K. Homidovb a Turin Polytechnic University, Tashkent, Uzbekistan
b National University of Uzbekistan named after Mirzo Ulugbek, Tashkent, Uzbekistan
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.
Received: 03.10.2022 Accepted: 10.05.2023
Citation:
A. A. Dzhalilov, M. K. Homidov, “Hitting functions for mixed partitions”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 33:2 (2023), 197–211
Linking options:
https://www.mathnet.ru/eng/vuu844 https://www.mathnet.ru/eng/vuu/v33/i2/p197
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Abstract page: | 101 | Full-text PDF : | 21 | References: | 20 |
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