Multiple ergodic averages for tempered functions

Andreas Koutsogiannis

doi:10.3934/dcds.2020314

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2021, Volume 41, Issue 3: 1177-1205. Doi: 10.3934/dcds.2020314

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Multiple ergodic averages for tempered functions

Andreas Koutsogiannis

The Ohio State University, Department of Mathematics, Columbus, Ohio, USA

Received: April 30, 2020

Revised: June 30, 2020

Early access: August 2020

Published: March 2021

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Abstract

Following Frantzikinakis' approach on averages for Hardy field functions of different growth, we add to the topic by studying the corresponding averages for tempered functions, a class which also contains functions that oscillate and is in general more restrictive to deal with. Our main result is the existence and the explicit expression of the $ L^2 $-norm limit of the aforementioned averages, which turns out, as in the Hardy field case, to be the "expected" one. The main ingredients are the use of, the now classical, PET induction (introduced by Bergelson), covering a more general case, namely a "nice" class of tempered functions (developed by Chu-Frantzikinakis-Host for polynomials and Frantzikinakis for Hardy field functions) and some equidistribution results on nilmanifolds (analogous to the ones of Frantzikinakis' for the Hardy field case).

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Mathematics Subject Classification: Primary: 37A30; Secondary: 37A05.

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