Uniform Diophantine approximation on the plane for $β$-dynamical systems

In this paper, we investigate the two-dimensional uniform Diophantine approximation in $β$-dynamical systems. Let βᵢ > 1(i=1,2) be real numbers, and let Tβᵢ denote the βᵢ-transformation defined on $[0, 1]$. For each (x, y) ∈[0,1]², we define the asymptotic approximation exponent vβ₁, β₂(x, y)= \0 ≤ v<∞: arrayl Tβ₁ⁿ x<β₁-n v \\ Tβ₂ⁿ y<β₂-n v array for infinitely many n ∈ N\ , and the uniform approximation exponent $$ v̂β_1, β_2(x, y)= \{0 ≤ v̂<∞: ∀~ N 1, ∃ 1 ≤ n ≤ N such that {array}{l} Tβ_1^n x < β_1-N v̂ \\ Tβ_2^n y < β_2-N v̂ {array}\} . $$ We calculate the Hausdorff dimension of the intersection $$\{(x, y) ∈[0,1]^2: v̂β_1, β_2(x, y)=v̂ and vβ_1, β_2(x, y)=v\}$$ for any $v̂$ and $v$ satisfying $log β_2{β_1}>{v̂}{v}(1+v)$. As a corollary, we establish a definite formula for the Hausdorff dimension of the level set of the uniform approximation exponent.