  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.
