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

We show that a continuous abelian action (in particular Rᵈ) on a compact metric space equipped with an invariant ergodic measure has discrete spectrum if and only it is $μ-$mean equicontinuous (proven for Zᵈ 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.