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.