higher-dimensional-shrinking-target-problem-for-beta-dynamical-systems.pdf (360.35 kB)

# Higher-Dimensional Shrinking Target Problem for Beta Dynamical Systems

Version 2 2023-09-08, 03:53

Version 1 2022-05-12, 05:10

journal contribution

posted on 2022-05-12, 05:10 authored by Mumtaz HussainMumtaz Hussain, WEILIANG WANGAbstract
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
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.