Let $([0,1], T_{\beta})$ be the beta-dynamical system for $\beta>1$. For any $x, y\in [0,1]$, write $d_n(x;y):=d(T^n_{\beta}(x), y)$ to measure the distance of the $n$-th orbit $T^n_{\beta}(x)$ of $x$ to the point $y$. This talk is devoted to investigating the size of following recurrence set and shrinking target problem $R(\psi, T_{\beta}):=\big\{x: d_n(x; x)<\psi(n, x), \text{i.o.}\ n\in \mathbb{N}\big\},$ $S(\psi, T_{\beta}, y):=\big\{x: d_n(x; y)<\psi(n, x), \text{i.o.}\ n\in \mathbb{N}\big\},$ where $\psi$ is some positive function given in advance. Among them, some algebraic and geometric properties shared by $\beta$-expansion are also investigated to serve for the main results.
