Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems

Felipe García-Ramos; Brian Marcus

doi:10.3934/dcds.2019030

Article Contents

2019, Volume 39, Issue 2: 729-746. Doi: 10.3934/dcds.2019030

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Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems

Felipe García-Ramos^, and
Brian Marcus

1.
CONACyT / Instituto de Física, Universidad Autónoma de San Luis Potosí (UASLP), Av. Manuel Nava #6, Zona Universitaria, San Luis Potosí, S.L.P., 78290, México
2.
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

* Corresponding author
* Corresponding author

Received: August 31, 2017

Revised: July 31, 2018

Published: January 2019

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We show that a continuous abelian action (in particular $\mathbb{R}^{d}$) on a compact metric space equipped with an invariant ergodic measure has discrete spectrum if and only it is $μ-$mean equicontinuous (proven for $\mathbb{Z}^{d}$ in [14]). In order to do this we introduce mean equicontinuity and mean sensitivity with respect to a function. We study this notion in the topological and measure theoretic setting. In the measure theoretic case we characterize almost periodic functions with these concepts and in the topological case we show that weakly almost periodic functions are mean equicontinuous (the converse does not hold). We compare our results with some results in the theory of Delone dynamical systems and quasicrystals.

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Mathematics Subject Classification: 37A05, 37A25, 37A35, 37B05.

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