HIGHER-DIMENSIONAL SHRINKING TARGET PROBLEM FOR BETA DYNAMICAL SYSTEMS

Abstract We consider the two-dimensional shrinking target problem in beta dynamical systems (for general $\beta>1$ ) with errors of approximation. Let $f, g$ be two positive continuous functions. For any $x_0,y_0\in [0,1]$ , define 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 <e^{-S_nf(x)}\\[1ex] T_{\beta}^{n}y-y_{0}\rvert < e^{-S_ng(y)} \end{array} \ {\text{for infinitely many}} n\in \mathbb N \right\}, \end{align*} where $S_nf(x)=\sum _{j=0}^{n-1}f(T_\beta ^jx)$ is Birkhoff sum. calculate Hausdorff dimension this and prove that it solution to some pressure function. This represents first result kind for higher-dimensional systems.