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arXiv 2208.08059math.DS

分段单调区间映射的联合遍历性

Joint ergodicity of piecewise monotone interval maps

Vitaly Bergelson, Younghwan Son

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AI总结：

本文研究分段单调区间映射的一致联合遍历性，建立方便判据并给出区间交换变换、β-变换与Gauss映射的联合遍历应用，同时证明斜帐篷映射和单位圆上有限Blaschke乘积限制的联合混合性。

摘要：

对于 $i = 0, 1, 2, \dots, k$，设 $μ_i$ 是 $[0,1]$ 上与 Lebesgue 测度 $λ$ 等价的 Borel 概率测度，并设 $T_i:[0,1] \rightarrow [0,1]$ 是保 $μ_i$ 的遍历变换。我们称变换 $T_0, T_1, \dots, T_k$ 关于 $(λ; μ_0, μ_1, \dots, μ_k)$ 是一致联合遍历的，如果对任意 $f_0, f_1, \dots, f_k \in L^{\infty}$，有 \\[ \lim\limits_{N -M \rightarrow \infty} \frac{1}{N-M } \sum\limits_{n=M}^{N-1} f_0 ( T_0^{n} x) \cdot f_1 (T_1^n x) \cdots f_k (T_k^n x) = \prod_{i=0}^k \int f_i \\, d μ_i \quad \text{在 } L^2(λ) \text{ 中成立。} \\] 我们建立了一致联合遍历性的方便判据，并获得了大量应用，其中大多数涉及区间映射。以下是对其中一个应用的描述。设 $T_G$ 表示 Gauss 映射，$T_G(x) = \frac{1}{x} \\, (\bmod \\, 1)$，并对 $β>1$，设 $T_β$ 表示由 $T_β x = βx \\, (\bmod \\,1)$ 定义的 $β$-变换。设 $T_0$ 是一个遍历的区间交换变换。设 $β_1 , \cdots , β_k$ 是互不相同的实数且 $β_i >1$，并假设对所有 $i = 1, 2, \dots, k$ 都有 $\log β_i \ne \frac{π^2}{6 \log 2}$。那么对任意 $f_{0}, f_1, f_{2}, \dots, f_{k+1} \in L^{\infty} (λ)$，有 \begin{equation*} \begin{split} \lim\limits_{N -M \rightarrow \infty} \frac{1}{N -M } \sum\limits_{n=M}^{N-1} & f_{0} (T_0^n x) \cdot f_{1} (T_{β_1}^n x) \cdots f_{k} (T_{β_k}^n x) \cdot f_{k+1} (T_G^n x) \\\\ &= \int f_{0} \\, d λ\cdot \prod_{i=1}^k \int f_{i} \\, d μ_{β_i} \cdot \int f_{k+1} \\, d μ_G \quad \text{在 } L^{2}(λ) \text{ 中成立。} \end{split} \end{equation*} 我们还研究了联合混合现象。除其他结果外，我们建立了斜帐篷映射以及有限 Blaschke 乘积在单位圆上的限制的联合混合性。

英文摘要：

For $i = 0, 1, 2, \dots, k$, let $μ_i$ be a Borel probability measure on $[0,1]$ which is equivalent to Lebesgue measure $λ$ and let $T_i:[0,1] \rightarrow [0,1]$ be $μ_i$-preserving ergodic transformations. We say that transformations $T_0, T_1, \dots, T_k$ are uniformly jointly ergodic with respect to $(λ; μ_0, μ_1, \dots, μ_k)$ if for any $f_0, f_1, \dots, f_k \in L^{\infty}$, \[ \lim\limits_{N -M \rightarrow \infty} \frac{1}{N-M } \sum\limits_{n=M}^{N-1} f_0 ( T_0^{n} x) \cdot f_1 (T_1^n x) \cdots f_k (T_k^n x) = \prod_{i=0}^k \int f_i \, d μ_i \quad \text{ in } L^2(λ). \] We establish convenient criteria for uniform joint ergodicity and obtain numerous applications, most of which deal with interval maps. Here is a description of one such application. Let $T_G$ denote the Gauss map, $T_G(x) = \frac{1}{x} \, (\bmod \, 1)$, and, for $β>1$, let $T_β$ denote the $β$-transformation defined by $T_β x = βx \, (\bmod \,1)$. Let $T_0$ be an ergodic interval exchange transformation. Let $β_1 , \cdots , β_k$ be distinct real numbers with $β_i >1$ and assume that $\log β_i \ne \frac{π^2}{6 \log 2}$ for all $i = 1, 2, \dots, k$. Then for any $f_{0}, f_1, f_{2}, \dots, f_{k+1} \in L^{\infty} (λ)$, \begin{equation*} \begin{split} \lim\limits_{N -M \rightarrow \infty} \frac{1}{N -M } \sum\limits_{n=M}^{N-1} & f_{0} (T_0^n x) \cdot f_{1} (T_{β_1}^n x) \cdots f_{k} (T_{β_k}^n x) \cdot f_{k+1} (T_G^n x) &= \int f_{0} \, d λ\cdot \prod_{i=1}^k \int f_{i} \, d μ_{β_i} \cdot \int f_{k+1} \, d μ_G \quad \text{in } L^{2}(λ). \end{split} \end{equation*} We also study the phenomenon of joint mixing. Among other things we establish joint mixing for skew tent maps and for restrictions of finite Blaschke products to the unit circle.

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