In this paper, we investigate the two-dimensional uniform Diophantine approximation in $\beta$-dynamical systems. Let $\beta_i > 1(i=1,2)$ be real numbers, and let $T_{ \beta_i}$ denote the $\beta_i$-transformation defined on $[0, 1]$. For each $(x, y) \in[0,1]^2$, we define the asymptotic approximation exponent $$ v_{ \beta_1, \beta_2}(x, y)=\sup \left\{ 0 \leq v<\infty: \begin{ array}{ l} T_{ \beta_1}^n x<\beta_1^{ -n v} \\ T_{ \beta_2}^n y<\beta_2^{ -n v} \end{ array} \text { for infinitely many } n \in \mathbb{ N}\right\} \text { , } $$ and the uniform approximation exponent $$ \hat{ v}_{ \beta_1, \beta_2}(x, y)=\sup \left\{ 0 \leq \hat{ v}<\infty: \forall~ N \gg 1, \exists 1 \leq n \leq N \text { such that } \begin{ array}{ l} T_{ \beta_1}^n x < \beta_1^{ -N \hat{ v}} \\ T_{ \beta_2}^n y < \beta_2^{ -N \hat{ v}} \end{ array}\right\} . $$ We calculate the Hausdorff dimension of the intersection $$\left\{ (x, y) \in[0,1]^2: \hat{ v}_{ \beta_1, \beta_2}(x, y)=\hat{ v} \text { and } v_{ \beta_1, \beta_2}(x, y)=v\right\}$$ for any $\hat{ v}$ and $v$ satisfying $\log _{ \beta_2}{ \beta_1}>\frac{ \hat{ v}}{ v}(1+v)$. As a corollary, we establish a definite formula for the Hausdorff dimension of the level set of the uniform approximation exponent.
Categories: math.DS