# American Institute of Mathematical Sciences

February  2019, 39(2): 729-746. doi: 10.3934/dcds.2019030

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

 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

Received  September 2017 Revised  August 2018 Published  November 2018

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.

Citation: Felipe García-Ramos, Brian Marcus. Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 729-746. doi: 10.3934/dcds.2019030
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