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Higher-Dimensional Shrinking Target Problem for Beta Dynamical Systems

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Version 2 2023-09-08, 03:53
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journal contribution
posted on 2023-09-08, 03:53 authored by Mumtaz HussainMumtaz Hussain, Weiliang Wang

We consider the two-dimensional shrinking target problem in beta dynamical systems (for general $\beta>1$ ) with general errors of approximation. Let $f, g$ be two positive continuous functions. For any $x_0,y_0\in [0,1]$ , define the shrinking target set $$ \begin{align*}E(T_\beta, f,g):=\left\{(x,y)\in [0,1]^2: \begin{array}{@{}ll@{}} \lvert T_{\beta}^{n}x-x_{0}\rvert where $S_nf(x)=\sum _{j=0}^{n-1}f(T_\beta ^jx)$ is the Birkhoff sum. We calculate the Hausdorff dimension of this set and prove that it is the solution to some pressure function. This represents the first result of this kind for the higher-dimensional beta dynamical systems.

Funding

The first-named author was supported by ARCDP200100994. The second author was supported by Natural Science Research Project of West Anhui University (No. WGKQ2021020) and Provincial Natural Science Research Project of Anhui Colleges (No. KJ2021A0950).

History

Publication Date

2023-06-01

Journal

Journal of the Australian Mathematical Society

Volume

114

Issue

3

Pagination

289 - 311

Publisher

Cambridge University Press

ISSN

1446-7887

Rights Statement

© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

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