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

Abstract

We show that an $R^d$-topological dynamical system equipped with an invariant ergodic measure has discrete spectrum if and only it is $\mu$-mean equicontinuous (proven for $Z^d$ before). 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 and in the topological case we show that weakly almost periodic functions are mean equicontinuous (the converse does not hold).

References in corpus (3)

• Pure Point Dynamical and Diffraction Spectra
• Mean equicontinuity and mean sensitivity
• An autocorrelation and discrete spectrum for dynamical systems on metric spaces

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• Local non-periodic order and diam-mean equicontinuity on cellular automata
• Amorphic complexity of group actions with applications to quasicrystals
• Diameter mean equicontinuity and cellular automata
• Discrete spectrum for amenable group actions
• Weak mean equicontinuity for a countable discrete amenable group action
• When is a dynamical system mean sensitive?