Deviation for interval exchange transformations
Deviation for interval exchange transformations
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DOI：
10.1017/s0143385797086215
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发表时间：
1997-12
影响因子：
0.9
通讯作者：
A. Zorich
中科院分区：
文献类型：
--
作者：
A. Zorich
考虑区间交换变换 $T$ 的一段长轨迹 $x, T(x), T(T(x)), \ldots, T^{n-1}(x)$。通用区间交换变换是唯一遍历的。因此，遍历定理预测我们的轨迹访问第 $i$ 个子区间的次数 $\chi_i(x,n)$ 大约为 $\lambda_i n$。这里$\lambda_i$是我们的单位区间$X$对应的子区间的长度。在本文中，我们给出了第 $i$ 个子区间 $X_i$ 的实际访问次数与遍历定理预测的偏差的估计。我们证明，对于几乎所有区间交换变换，以下界限都是有效的： $$ \max_{\ssty x\in X \atop \ssty 1\le i\le m} \limsup_{n\to +\infty} \frac {\log | \chi_i(x,n) -\lambda_in|}{\log n} = \frac{\theta_2}{\theta_1} 0$ 对于所有 $n\ge N(\varepsilon)$，对数比将小于 $\theta_2(\pi)/\theta_1(\pi)+\varepsilon $，其中 $N(\varepsilon)$ 不依赖于起点 $x\in X$。
Consider a long piece of a trajectory $x, T(x), T(T(x)), \ldots, T^{n-1}(x)$ of an interval exchange transformation $T$. A generic interval exchange transformation is uniquely ergodic. Hence, the ergodic theorem predicts that the number $\chi_i(x,n)$ of visits of our trajectory to the $i$th subinterval would be approximately $\lambda_i n$. Here $\lambda_i$ is the length of the corresponding subinterval of our unit interval $X$. In this paper we give an estimate for the deviation of the actual number of visits to the $i$th subinterval $X_i$ from one predicted by the ergodic theorem. We prove that for almost all interval exchange transformations the following bound is valid: $$ \max_{\ssty x\in X \atop \ssty 1\le i\le m} \limsup_{n\to +\infty} \frac {\log | \chi_i(x,n) -\lambda_in|}{\log n} = \frac{\theta_2}{\theta_1} 0$ the ratio of logarithms would be less than $\theta_2(\pi)/\theta_1(\pi)+\varepsilon $ for all $n\ge N(\varepsilon)$, where $N(\varepsilon)$ does not depend on the starting point $x\in X$.