MATHEMATICS

Hitting functions for mixed partitions

A. A. Dzhalilov^a, M. K. Homidov<sup>b

a</sup> 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.

UDC: 517.938

MSC: 37C05, 37C15, 37E05, 37E10, 37E20, 37B10

Received: 03.10.2022
Accepted: 10.05.2023

Language: English

DOI: 10.35634/vm230201

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