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

Felipe García-Ramos, Brian Marcus

Published 2015-09-17Version 1

We define forms of mean sensitivity for measure theoretical and topological dynamical systems with respect to a given function f. In the measure theoretic case, we show that mean sensitivity with respect to f is complementary to almost periodicity of f, and we obtain a new dynamical characterization of continuous spectrum for Z^d and R^d ergodic systems. In the topological case, we show that a topological version of mean sensitivity with respect to a continuous f is complementary to a topological notion of mean equicontinuity with respect to f. In the hybrid topological/measure-theoretic case, we show that a hybrid notion of mean equicontnuity for f is complementary to the measure-theoretic notion of mean sensitivity.