Convergence of polynomial ergodic averages
The main result of the paper is the proof of \(L^2\)-convergence of a product of measurable functions evaluated along polynomial times. Let \((X,{\mathcal B}, \mu,T)\) be an invertible dynamical system, let \(\{p_i(n)\}_{i=1}^\ell\) be integer polynomials, and let at least one of the following conditions hold: (i) the system \((X,{\mathcal B}, \mu,T)\) is totally ergodic; (ii) the polynomials are all of degree greater than one; (iii) the polynomials are all of degree one. Then for any collection of \(L^\infty(\mu)\) functions \(\{f_i\}_{i=1}^\ell\) and any two sequences \(\{M_i\}, \{N_i\}\) of integers with \(N_i\to\infty\) the averages \[ \frac1{N_i} \sum_{n=M_i}^{M_i+N_i-1} \prod_{k=1}^l f_k\circ T^{p_k} ~d\mu \] converge as \(i\to\infty\).
scientific article; zbMATH DE number 5003388
scientific article