In this paper, we investigate the two-dimensional shrinking target problem in beta-dynamical systems. Let β > 1 be a real number and define the β-transformation on [0, 1] by T β : x → βx mod 1. Let Ψ i (i = 1, 2) be two positive functions on N such that Ψ i → 0 when n → ∞. We determine the Lebesgue measure and Hausdorff dimension for the lim sup set2010 Mathematics subject classification: primary 11K55; secondary 11J83, 28A80, 37F35.
Let
$\unicode[STIX]{x1D6FD}>1$
be a real number and define the
$\unicode[STIX]{x1D6FD}$
-transformation on
$[0,1]$
by
$T_{\unicode[STIX]{x1D6FD}}:x\mapsto \unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$
. Let
$f:[0,1]\rightarrow [0,1]$
and
$g:[0,1]\rightarrow [0,1]$
be two Lipschitz functions. The main result of the paper is the determination of the Hausdorff dimension of the set
$$\begin{eqnarray}W(f,g,\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2})=\big\{(x,y)\in [0,1]^{2}:|T_{\unicode[STIX]{x1D6FD}}^{n}x-f(x)|<\unicode[STIX]{x1D6FD}^{-n\unicode[STIX]{x1D70F}_{1}(x)},|T_{\unicode[STIX]{x1D6FD}}^{n}y-g(y)|<\unicode[STIX]{x1D6FD}^{-n\unicode[STIX]{x1D70F}_{2}(y)}~\text{for infinitely many}~n\in \mathbb{N}\big\},\end{eqnarray}$$
where
$\unicode[STIX]{x1D70F}_{1}$
,
$\unicode[STIX]{x1D70F}_{2}$
are two positive continuous functions with
$\unicode[STIX]{x1D70F}_{1}(x)\leq \unicode[STIX]{x1D70F}_{2}(y)$
for all
$x,y\in [0,1]$
.
Let
$0\leq \alpha \leq \infty $
,
$0\leq a\leq b\leq \infty $
and
$\psi $
be a positive function defined on
$(0,\infty )$
. This paper is concerned with the growth of
$L_{n}(x)$
, the largest digit of the first n terms in the Lüroth expansion of
$x\in (0,1]$
. Under some suitable assumptions on the function
$\psi $
, we completely determine the Hausdorff dimensions of the sets
$$\begin{align*}E_\psi(\alpha)=\bigg\{x\in(0,1]: \lim\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=\alpha\bigg\} \end{align*}$$
and
$$\begin{align*}E_\psi(a,b)=\bigg\{x\in(0,1]: \liminf\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=a, \limsup\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=b\bigg\}. \end{align*}$$
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