| 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. |