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We consider the two dimensional shrinking target problem in the betadynamical system for general $\beta>1$ and with the general error ofapproximations. 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 calculatethe Hausdorff dimension of this set and prove that it is the solution to somepressure function. This represents the first result of this kind for the higherdimensional beta dynamical systems.

Higher dimensional shrinking target problem in beta dynamical systems