Transitive Systems with Zero Sequence Entropy and Sequence Entropy Pairs

Abstract

A measure-preserving transformation (resp. a topological system) is null if the metric (resp. topological) sequence entropy is zero for any sequence. Kushnirenko showed that an ergodic measure-preserving transformation T has discrete spectrum if and only if it is null. We prove that for a minimal system the above statement remains true modulo an almost one-to-one extension, i.e. if a minimal system (X,T) is null, then (X,T) is an almost one-to-one extension of an equicontinuous system. It allows us to show that a scattering system is disjoint from any null minimal system. Moreover, we show that if a transitive non-minimal system (X,T) is null then there are non-empty open sets U and V of X such that N(U,V) has zero upper Banach density. Examples of null minimal systems which are not equicontinuous exist.

Localizing the notion of sequence entropy, we define sequence entropy pairs and show that there is a maximal null factor for any system. Meanwhile, we define a weaker notion, namely weak mixing pairs. It turns out that a system is weakly mixing if and only if any pair not in the diagonal is a sequence entropy pair if and only if the same holds for a weak mixing pair, answering an open question by Blanchard, Host and Maass.

For a group action we show that the factor induced by the smallest invariant equivalence relation containing weak mixing pairs is equicontinuous, supplying another proof concerning regionally proximal relation. Furthermore, for a minimal distal system the set of sequence entropy pairs coincides with the regionally proximal relation and thus a non-equicontinuous minimal distal system is not null.