Pointwise convergence of some multiple ergodic averages
| Autor: | Sebastián Donoso, Wenbo Sun |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Pointwise convergence
Ergodic system Mathematics::Number Theory General Mathematics 010102 general mathematics Mathematics::General Topology Dynamical Systems (math.DS) 37A30 54H20 01 natural sciences Combinatorics 010104 statistics & probability FOS: Mathematics Ergodic theory Mathematics - Dynamical Systems 0101 mathematics Mathematics |
| Zdroj: | Advances in Mathematics. 330:946-996 |
| ISSN: | 0001-8708 |
| DOI: | 10.1016/j.aim.2018.03.022 |
| Popis: | We show that for every ergodic system ( X , μ , T 1 , … , T d ) with commuting transformations, the average 1 N d + 1 ∑ 0 ≤ n 1 , … , n d ≤ N − 1 ∑ 0 ≤ n ≤ N − 1 f 1 ( T 1 n ∏ j = 1 d T j n j x ) f 2 ( T 2 n ∏ j = 1 d T j n j x ) ⋯ f d ( T d n ∏ j = 1 d T j n j x ) converges for μ-a.e. x ∈ X as N → ∞ . If X is distal, we prove that the average 1 N ∑ n = 0 N − 1 f 1 ( T 1 n x ) f 2 ( T 2 n x ) ⋯ f d ( T d n x ) converges for μ-a.e. x ∈ X as N → ∞ . We also establish the pointwise convergence of averages along cubical configurations arising from a system with commuting transformations. Our methods combine the existence of sated and magic extensions introduced by Austin and Host respectively with ideas on topological models by Huang, Shao and Ye. |
| Databáze: | OpenAIRE |
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