Abstract:: In this talk we present recent developments and new results in the study of mean equicontinuity and weak mean equicontinuity in the context of countable discrete amenable groups, such as a characterization in terms of spectral theory. It is known that the notions of mean eq. and weak mean eq. do not depend on the process of averaging (the Følner sequence), whenever the action is minimal, or the acting group is Abelian. However, we present an example of a (non-minimal, but transitive) action where mean equicontinuity and weak mean equicontinuity do depend on the process of averaging. This observation leads to the new notions of essential mean equicontinuity and essential weak mean equicontinuity, for which we present characterizations.
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