论文标题
$ \ times a $ and $ \ times b $经验措施，不规则套装和熵
$\times a$ and $\times b$ empirical measures, the irregular set and entropy
论文作者
论文摘要
对于整数$ a $和$ b \ geq 2 $，让$ t_a $和$ t_b $ be乘以$ a $和$ b $ oN $ \ mathbb {t} = \ mathbb {r}/\ mathbb {z z} $。 $ t_a $和$ t_b $在$ \ mathbb {t} $上的操作称为$ \ times a，\ times b $动作，并且众所周知，如果$ a $ a和$ b $是多重独立的，则唯一的$ \ times a，\ times b $ b $ bb $不变且具有$ t_a $ t_a $ t_bes $ t_b的正面量度是$ t_b is t_b is te_b is pebes is pebes is pebes is pebes is pebes是pe e n e le pe e n e le peb is pe。但是，是否存在非平凡的$ \ times a，\ times b $不变和千古的度量。在本文中，我们研究了相对于$ \ times a，\ times b $的$ x \ in \ mathbb {t} $的经验度量不变度的度量为Hausdorff尺寸零。此外，我们在一组Hausdorff Dimension Zere的补充中获得了$ \ times a，\ times b $ o轨道的一些等分分配结果。
For integers $a$ and $b\geq 2$, let $T_a$ and $T_b$ be multiplication by $a$ and $b$ on $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. The action on $\mathbb{T}$ by $T_a$ and $T_b$ is called $\times a,\times b$ action and it is known that, if $a$ and $b$ are multiplicatively independent, then the only $\times a,\times b$ invariant and ergodic measure with positive entropy of $T_a$ or $T_b$ is the Lebesgue measure. However, whether there exists a nontrivial $\times a,\times b$ invariant and ergodic measure is not known. In this paper, we study the empirical measures of $x\in\mathbb{T}$ with respect to the $\times a,\times b$ action and show that the set of $x$ such that the empirical measures of $x$ do not converge to any measure has Hausdorff dimension $1$ and the set of $x$ such that the empirical measures can approach a nontrivial $\times a,\times b$ invariant measure has Hausdorff dimension zero. Furthermore, we obtain some equidistribution result about the $\times a,\times b$ orbit of $x$ in the complement of a set of Hausdorff dimension zero.
