Joint ergodicity of piecewise monotone interval maps

Abstract

For $i = 0, 1, 2, \dots, k$, let $\mu_i$ be a Borel probability measure on $[0,1]$ which is equivalent to Lebesgue measure $\lambda$ and let $T_i:[0,1] \rightarrow [0,1]$ be $\mu_i$-preserving ergodic transformations. We say that transformations $T_0, T_1, \dots, T_k$ are uniformly jointly ergodic with respect to $(\lambda; \mu_0, \mu_1, \dots, \mu_k)$ if for any $f_0, f_1, \dots, f_k \in L^{\infty}$, \[ \lim\limits_{N -M \rightarrow \infty} \frac{1}{N-M } \sum\limits_{n=M}^{N-1} f_0 ( T_0^{n} x) \cdot f_1 (T_1^n x) \cdots f_k (T_k^n x) = \prod_{i=0}^k \int f_i \, d \mu_i \quad \text{ in } L^2(\lambda). \] 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 $\beta >1$, let $T_{\beta}$ denote the $\beta$-transformation defined by $T_{\beta} x = \beta x \, (\bmod \,1)$. Let $T_0$ be an ergodic interval exchange transformation. Let $\beta_1 , \cdots , \beta_k$ be distinct real numbers with $\beta_i >1$ and assume that $\log \beta_i \ne \frac{\pi^2}{6 \log 2}$ for all $i = 1, 2, \dots, k$. Then for any $f_{0}, f_1, f_{2}, \dots, f_{k+1} \in L^{\infty} (\lambda)$, \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_{\beta_1}^n x) \cdots f_{k} (T_{\beta_k}^n x) \cdot f_{k+1} (T_G^n x) &= \int f_{0} \, d \lambda \cdot \prod_{i=1}^k \int f_{i} \, d \mu_{\beta_i} \cdot \int f_{k+1} \, d \mu_G \quad \text{in } L^{2}(\lambda). \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.