Joint ergodicity of piecewise monotone interval maps
arXiv:2208.08059 · doi:10.1088/1361-6544/acd29a
Abstract
For , let be a Borel probability measure on which is equivalent to Lebesgue measure and let be -preserving ergodic transformations. We say that transformations are uniformly jointly ergodic with respect to if for any , \[ \lim\limits_{N -M \rightarrow \infty} \frac{1}{N-M } \sum\limits_{n=M}^{N-1} f_0 ( T_0^{n} x) \cdot f_1 (T_1^n x) \cdots f_k (T_k^n x) = \prod_{i=0}^k \int f_i \, d μ_i \quad \text{ in } L^2(λ). \] 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 denote the Gauss map, , and, for , let denote the -transformation defined by . Let be an ergodic interval exchange transformation. Let be distinct real numbers with and assume that for all . Then for any , \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.
38 pages