论文标题
Beta动力学系统中的不均匀和同时双苯胺近似
Inhomogeneous and simultaneous Diophantine approximation in beta dynamical systems
论文作者
论文摘要
在本文中，我们研究了Beta动力学系统中的不均匀和同时进行双磷酸的近似。对于$β> 1 $，让$t_β$是$ [0,1] $上的$β$ - 转化。我们确定集合\ [\ left \ {（x，y）\ in [0,1]^2：|t_β^nx-f（x，x，y）| <φ（x，y） [0,1] $是lipschitz函数，$φ$是$ \ mathbb {n} $的正函数。令$β_2\ geqβ_1> 1 $，$ f_1，f_1：[0,1] \ to [0,1] $是两个Lipschitz函数，$τ_1，τ_2$是$ [0,1] $的两个正面连续功能。我们还确定集合的Hausdorff尺寸\ [\ left \ {（x，y）\ in [0,1]^2：\ begin {aligned}＆| t_ {β_1}^nx-f_1（x）和| \ [\ left \ {（x，y）\ in [0,1]^2：\ begin {Aligned}＆| t_ {β_1}^nx-g_1（x，x，y）|<β_1^{ - nτ_1（x） ＆| t_ {β_2}^ny-g_2（x，y）| <β_2^{ - nτ_2（y）} \ end {aligned} \ text {for Ind} n \ in \ mathbb {n}是两个Lipschitz功能。
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.
