# Higher dimensional shrinking target problem in beta dynamical systems

## TL;DR

This paper investigates the higher-dimensional shrinking target problem in beta dynamical systems, calculating the Hausdorff dimension of the set of points that approximate a target infinitely often, extending prior one-dimensional results.

## Contribution

It provides the first Hausdorff dimension calculation for higher-dimensional beta dynamical systems' shrinking target sets, using pressure functions.

## Key findings

- Hausdorff dimension characterized by a pressure function
- First such result for higher-dimensional beta systems
- Extends shrinking target problem to two dimensions

## Abstract

We consider the two dimensional shrinking target problem in the beta dynamical system for general $\beta>1$ and with the general error of approximations. Let $f, g$ be two positive continuous functions. For any $x_0,y_0\in[0,1]$, define the shrinking target set $$ E(T_\beta, f,g):=\left\{(x,y)\in [0,1]^2: \begin{array}{ll} |T_{\beta}^{n}x-x_{0}|<e^{-S_nf(x)}\\ [1ex] |T_{\beta}^{n}y-y_{0}|< e^{-S_ng(y)} \end{array} \ {\text{for infinitely many}} \ n\in \N \right\}, $$ 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.