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Pointwise multiple averages for sublinear functions

Part of: Ergodic theory Classical measure theory

Published online by Cambridge University Press:  05 November 2018

SEBASTIÁN DONOSO ,
ANDREAS KOUTSOGIANNIS  and
WENBO SUN
SEBASTIÁN DONOSO
Affiliation:
Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins, Av. Libertador Bernardo O’Higgins 611, Rancagua, 2841959, Chile email [email protected]
ANDREAS KOUTSOGIANNIS
Affiliation:
Department of Mathematics,The Ohio State University, 231 West 18th Avenue, Columbus, OH 43210-1174, USA email [email protected], [email protected]
WENBO SUN
Affiliation:
Department of Mathematics,The Ohio State University, 231 West 18th Avenue, Columbus, OH 43210-1174, USA email [email protected], [email protected]
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Abstract

For any measure-preserving system $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T_{1},\ldots ,T_{d})$ with no commutativity assumptions on the transformations $T_{i},$$1\leq i\leq d,$ we study the pointwise convergence of multiple ergodic averages with iterates of different growth coming from a large class of sublinear functions. This class properly contains important subclasses of Hardy field functions of order zero and of Fejér functions, i.e., tempered functions of order zero. We show that the convergence of the single average, via an invariant property, implies the convergence of the multiple one. We also provide examples of sublinear functions which are, in general, bad for convergence on arbitrary systems, but good for uniquely ergodic systems. The case where the fastest function is linear is addressed as well, and we provide, in all the cases, an explicit formula of the limit function.

Keywords

pointwise convergencemultiple averagessublinear functionsFejér functionsHardy functions

MSC classification

Primary: 37A05: Measure-preserving transformations
Secondary: 37A30: Ergodic theorems, spectral theory, Markov operators 28A99: None of the above, but in this section
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 6 , June 2020 , pp. 1594 - 1618
Copyright
© Cambridge University Press, 2018

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