We define weaker forms of topological and measure theoretical equicontinuity for topological dynamical systems, and we study their relationships with sequence entropy and systems with discrete spectrum.We show that for topological systems equipped with ergodic measures having discrete spectrum is equivalent to µ-mean equicontinuity.In the purely topological category we show that minimal subshifts with zero topological sequence entropy are strictly contained in the diam-mean equicontinuous systems; and that transitive almost automorphic subshifts are diam-mean equicontinuous if and only if they are regular (i.e. the maximal equicontinuous factor map is 1-1 on a set of full Haar measure).For both categories we find characterizations using stronger versions of the classical notion of sensitivity. As a consequence we obtain a dichotomy between discrete spectrum and a strong form of measure theoretic sensitivity.

We show that an R d −topological dynamical system equipped with an invariant ergodic measure has discrete spectrum if and only it is µ−mean equicontinuous (proven for Z d in [16]). 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). We compare our results with results in the theory of Delone dynamical systems and quasicrystals.

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We provide a unifying approach which links results on algebraic actions by Lind and Schmidt, Chung and Li, and a topological result by Meyerovitch that relates entropy to the set of asymptotic pairs. In order to do this we introduce a series of Markovian properties and, under the assumption that they are satisfied, we prove several results that relate topological entropy and asymptotic pairs (the homoclinic group in the algebraic case). As new applications of our method, we give a characterization of the homoclinic group of any finitely presented expansive algebraic action of (1) any elementary amenable, torsion-free group or (2) any left orderable amenable group, using the language of independence entropy pairs.

This article is devoted to study which conditions imply that a topological dynamical system is mean sensitive and which do not. Among other things we show that every uniquely ergodic, mixing system with positive entropy is mean sensitive. On the other hand we provide an example of a transitive system which is cofinitely sensitive or Devaney chaotic with positive entropy but fails to be mean sensitive.As applications of our theory and examples, we negatively answer an open question regarding equicontinuity/sensitivity dichotomies raised by Tu, we introduce and present results of locally mean equicontinuous systems and we show that mean sensitivity of the induced hyperspace does not imply that of the phase space.2010 Mathematics Subject Classification. 54H20, 37B10, 54B20, 37B05.