1 results
TWO-DIMENSIONAL SHRINKING TARGET PROBLEM IN BETA-DYNAMICAL SYSTEMS
- Part of
-
- Journal:
- Bulletin of the Australian Mathematical Society / Volume 97 / Issue 1 / February 2018
- Published online by Cambridge University Press:
- 02 November 2017, pp. 33-42
- Print publication:
- February 2018
-
- Article
-
- You have access
- Export citation
-
In this paper, we investigate the two-dimensional shrinking target problem in beta-dynamical systems. Let
$\unicode[STIX]{x1D6FD}>1$ be a real number and define the 
$\unicode[STIX]{x1D6FD}$-transformation on 
$[0,1]$ by 
$T_{\unicode[STIX]{x1D6FD}}:x\rightarrow \unicode[STIX]{x1D6FD}x\;\text{mod}\;1$. Let 
$\unicode[STIX]{x1D6F9}_{i}$ (
$i=1,2$) be two positive functions on 
$\mathbb{N}$ such that 
$\unicode[STIX]{x1D6F9}_{i}\rightarrow 0$ when 
$n\rightarrow \infty$. We determine the Lebesgue measure and Hausdorff dimension for the 
$\limsup$ set for any fixed
$$\begin{eqnarray}W(T_{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6F9}_{1},\unicode[STIX]{x1D6F9}_{2})=\{(x,y)\in [0,1]^{2}:|T_{\unicode[STIX]{x1D6FD}}^{n}x-x_{0}|<\unicode[STIX]{x1D6F9}_{1}(n),|T_{\unicode[STIX]{x1D6FD}}^{n}y-y_{0}|<\unicode[STIX]{x1D6F9}_{2}(n)\text{ for infinitely many }n\in \mathbb{N}\}\end{eqnarray}$$
$x_{0},y_{0}\in [0,1]$.
