论文标题
分段单调间隔图的关节奇迹性
Joint ergodicity of piecewise monotone interval maps
论文作者
论文摘要
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. 我们说转换$ t_0，t_1，\ dots，t_k $对于$（λ；μ_0，μ_1，\ dots，μ_k）$均匀地共同共同ergodic $ \ lim \ limits_ {n -m \ rightarrow \ infty} \ frac {1} {n-m} {n-m} \ sum \ limits_ {n = m}^{n-1} f_0（t_0^{n} {n} x） \ prod_ {i = 0}^k \ int f_i \，dμ_i\ quad \ text {in} l^2（λ）。 \] 我们建立了统一关节性牙的方便标准，并获得了许多应用，其中大多数涉及间隔图。这是一个这样的应用程序的描述。令$ t_g $表示高斯地图，$ t_g（x）= \ frac {1} {x} \，（\ bmod \，1）$，对于$β> 1 $，令$t_β$表示$β$ - $β$ - 转移由$t_βx =βx=βx\ bmod，\ bmod，令$ t_0 $为一个千古的间隔交换转换。令$β_1，\ cdots，β_k$是$β_I> 1 $的不同实数，并假设所有$ i = 1、2，\ dots，k $，$ \logβ_i\ ne \ ne \ frac {π^2} {6 \ log 2} $。然后，对于任何$ f_ {0}，f_1，f_ {2}，\ dots，f_ {k+1} \ in l^{\ infty}（λ）$，\ begin {equination*} \ begin*} \ begin {split {splite} \ sum \ limits_ {n = m}^{n-1}＆f_ {0}（t_0^n x）\ cdot f_ {1}（t_ {β_1}^n x）\ cdots f_ {k {k}（t_ ＆= \ int f_ {0} \，dλ\ cdot \ prod_ {i = 1}^k \ int f_ {i} \，dμ_{β_i} \ cdot \ cdot \ cdot \ cdot \ cdot \ cdot \ int f_ {k+1} \ end {split} \ end {equation*} 我们还研究关节混合的现象。除其他事项外，我们还建立了偏斜帐篷地图的关节混合，并限制了有限的蓝皮产品对单位圆圈的限制。
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.
