Maximum rate of norm convergence in the ergodic theorem for groups Z^d and R^d
Maximum rate of norm convergence in the ergodic theorem for groups $\mathbb{Z}^d$ and $\mathbb{R}^d$
Сибирские электронные математические известия, 22, № 1, стр. 67-83 (2024)
Аннотация:
For $d$ pairwise commuting automorphisms (flows) of a probability space, ergodic averages over parallelepipeds are considered. It is shown that the maximum rate of their convergence in the $L_p$-norm is ${\mathcal{O}(\frac{1}{t_1t_2\cdots t_d}) }.$ A spectral criterion is also obtained for the maximum convergence rate in the $L_2$-norm.
Ключевые слова: rates of convergence in ergodic theorems, spectral measure, coboundaries, bundle of hyperplanes.