Uniform Diophantine approximation on the plane for ${\hat{I}}^2$-dynamical systems

Abstract

In this paper, we investigate the two-dimensional uniform Diophantine approximation in ${\hat{I}}^2$-dynamical systems. Let ${\hat{I}}_i^2>1(i=1,2)$ be real numbers, and let $T_{{\hat{I}}_i^2}$ denote the ${\hat{I}}_i^2$-transformation defined on $[0,1]$. For each $(x,y)\in [0,1{]}^2$, we define the asymptotic approximation exponent $$ v_{β_1, β_2}(x, y)=\sup \left\{0 \leq v<\infty: \begin{array}{l} T_{β_1}^n x<β_1^{-n v} \\ T_{β_2}^n y<β_2^{-n v} \end{array} \text { for infinitely many } n \in \mathbb{N}\right\} \text {, } $$ and the uniform approximation exponent $$ \hat{v}_{β_1, β_2}(x, y)=\sup \left\{0 \leq \hat{v}<\infty: \forall~ N \gg 1, \exists 1 \leq n \leq N \text { such that } \begin{array}{l} T_{β_1}^n x < β_1^{-N \hat{v}} \\ T_{β_2}^n y < β_2^{-N \hat{v}} \end{array}\right\} . $$ We calculate the Hausdorff dimension of the intersection $$\left\{(x,y)\in [0,1{]}^2:{\hat{v}}_{{\hat{I}}_1^2,{\hat{I}}_2^2}(x,y)=\hat{v}\text{ and }{v}_{{\hat{I}}_1^2,{\hat{I}}_2^2}(x,y)=v\right\}$$ for any $\hat{v}$ and $v$ satisfying ${\log }_{{\hat{I}}_2^2}{\hat{I}}_1^2>\frac{\hat{v}}{v}(1+v)$. As a corollary, we establish a definite formula for the Hausdorff dimension of the level set of the uniform approximation exponent.