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 $\mathbb{R}^d$

Kachurovskii A. G., Podvigin I. V., Khakimbaev A. J.
Siberian Electronic Mathematical Reports, 22, No. 1, pp. 67-83 (2025)
Full-text

УДК 517.987  
DOI: 10.33048/semi.2025.22.006  
MSC 37A30, 37A15

Abstract:

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.

 

Keywords: rates of convergence in ergodic theorems, spectral measure, coboundaries, bundle of hyperplanes.