Higher-Dimensional Shrinking Target Problem for Beta Dynamical Systems

Abstract

We consider the two-dimensional shrinking target problem in beta dynamical systems (for general $\beta>1$ ) with general errors of approximation. Let $f, g$ be two positive continuous functions. For any $x_0,y_0\in [0,1]$ , define the shrinking target set $$ \begin{align*}E(T_\beta, f,g):=\left\{(x,y)\in [0,1]^2: \begin{array}{@{}ll@{}} \lvert T_{\beta}^{n}x-x_{0}\rvert <e^{-S_nf(x)}\\[1ex] \lvert T_{\beta}^{n}y-y_{0}\rvert < e^{-S_ng(y)} \end{array} \ {\text{for infinitely many}} \ n\in \mathbb N \right\}, \end{align*} $$ where $S_nf(x)=\sum _{j=0}^{n-1}f(T_\beta ^jx)$ is the Birkhoff sum. We calculate the Hausdorff dimension of this set and prove that it is the solution to some pressure function. This represents the first result of this kind for the higher-dimensional beta dynamical systems.

Path-dependent shrinking target problems in beta-dynamical systems

“…Here P(s, T β1 , T β2 , f, g) is the modified pressure function defined through singular value function associated to the continuous functions f and g. When β 1 = β 2 and f ⩾ g, P(s, T β1 , T β1 , f, g) is consistent with the original one defined in [6]. The rigorous definitions of the singular value function and the modified pressure function require more preliminaries and will make introductory section a bit lengthy, so we postpone them to section 3.…”

“…In the current work, we appeal to the theory of large intersection property introduced by Falconer [3], and prove that the set E(T β1 , T β2 , f, g) has large intersections under the condition β 1 e min f(x) ⩾ β 2 , which has not been proved in any of the previous papers [1,6]. This means that the set E(T β1 , T β2 , f, g) belongs, for some 0 < s ⩽ 2, to the class G s ([0, 1] 2 ) of G δ -sets, with the property that any countable intersection of bi-Lipschitz images of sets in G s ([0, 1] 2 ) has Hausdorff dimension at least s. In particular, the Hausdorff dimension of E(T β1 , T β2 , f, g) is at least s. See the paper [3] for more details about those classes of sets.…”

“…Motivated by the weighted theory of classical Diophantine approximation, Hussain and Wang [6] extended the result of β-transformation by Bugeaud and Wang [1] to simultaneous setting. More precisely, let f, g be two positive continuous functions on [0, 1].…”

“…The set E(T β1 , T β2 , f, g) is the set of all points (x, y) in the unit square such that the pair (T n β1 x, T n β2 y) hits the shrinking rectangle B(x 0 , e −S β 1 n f(x) ) × B(y 0 , e −S β 2 n g(y) ) infinitely many times, where B(x, r) denotes the ball centered at x with radius r. When β 1 = β 2 , Hussain and Wang [6] obtained the Hausdorff dimension of the set E(T β1 , T β1 , f, g) under an additional assumption f ⩾ g.…”