本文研究β动力系统中的非齐次同时丢番图逼近。对于$β>1$，让$T_β$是$[0,1]$上的$β$ -转换。我们确定了集合\[\left\{(x,y)\in [0,1]^2: |T_β^nx-f(x,y)|<φ(n)\text{ for infinitely many }n\in\mathbb{N}\right\},\]的Lebesgue测度和Hausdorff维数，其中$f:[0,1]^2\to [0,1]$是Lipschitz函数，$φ$是$\mathbb{N}$上的正函数.设$β_2\geq β_1>1$，$f_1,f_2:[0,1]\to [0,1]$是两个Lipschitz函数，$τ_1,τ_2$是$[0,1]$上两个正连续函数.我们还确定了集合\[\left\{(x,y)\in [0,1]^2: \begin{aligned}&|T_{β_1}^nx-f_1(x)|<β_1^{-nτ_1(x)}\\ &|T_{β_2}^ny-f_2(y)|<β_2^{-nτ_2(y)}\end{aligned}\text{ for infinitely many }n\in\mathbb{N}\right\}.\]的Hausdorff维数在某些附加的假设下，还确定了集合\[\left\{(x,y)\in [0,1]^2: \begin{aligned}&|T_{β_1}^nx-g_1(x,y)|<β_1^{-nτ_1(x)}\\ &|T_{β_2}^ny-g_2(x,y)|<β_2^{-nτ_2(y)}\end{aligned}\text{ for infinitely many }n\in\mathbb{N}\right\}\]的Hausdorff维数，其中$g_1,g_2:[0,1]^2\to [0,1]$是两个Lipschitz函数.
英文摘要：
In this paper, we investigate inhomogeneous and simultaneous Diophantine approximation in beta dynamical systems. For $β>1$ let $T_β$ be the $β$-transformation on $[0,1]$. We determine the Lebesgue measure and Hausdorff dimension of the set \[\left\{(x,y)\in [0,1]^2: |T_β^nx-f(x,y)|<φ(n)\text{ for infinitely many }n\in\mathbb{N}\right\},\] where $f:[0,1]^2\to [0,1]$ is a Lipschitz function and $φ$ is a positive function on $\mathbb{N}$. Let $β_2\geq β_1>1$, $f_1,f_2:[0,1]\to [0,1]$ be two Lipschitz functions, $τ_1,τ_2$ be two positive continuous functions on $[0,1]$. We also determine the Hausdorff dimension of the set \[\left\{(x,y)\in [0,1]^2: \begin{aligned}&|T_{β_1}^nx-f_1(x)|<β_1^{-nτ_1(x)}\\ &|T_{β_2}^ny-f_2(y)|<β_2^{-nτ_2(y)}\end{aligned}\text{ for infinitely many }n\in\mathbb{N}\right\}.\] Under certain additional assumptions, the Hausdorff dimension of the set \[\left\{(x,y)\in [0,1]^2: \begin{aligned}&|T_{β_1}^nx-g_1(x,y)|<β_1^{-nτ_1(x)}\\ &|T_{β_2}^ny-g_2(x,y)|<β_2^{-nτ_2(y)}\end{aligned}\text{ for infinitely many }n\in\mathbb{N}\right\}\] is also determined, where $g_1,g_2:[0,1]^2\to [0,1]$ are two Lipschitz functions.
