This is a paper dealing with the local theory of topological entropy for dynamical systems given by a compact metric space and a homeomorphism .
Recall that the entropy pairs as markers of topological entropy were introduced in F. Blanchard [Bull. Soc. Math. Fr. 121, No. 4, 465–478 (1993; Zbl 0814.54027)] and the entropy tuples were defined in W. Huang and X. Ye [Isr. J. Math. 151, 237–279 (2006; Zbl 1122.37013)]. In D. Dou, X. Ye and G. Zhang [Nonlinearity 19, No. 1, 53–74 (2006; Zbl 1102.37007)] the notion of the entropy set was introduced. A closed set is an entropy set if and only if any tuple of elements of which are not all identical is an entropy tuple. An analogous result holds for -entropy sets where is an invariant measure.
In the paper under review the authors study the families and of entropy sets and -entropy sets and their closures and in the hyperspace . The sets and consist only of singletons. Denote by resp. the topological entropy (resp. measure-theoretical entropy with respect to the measure ) of . If (resp. ) then the action induced by on (resp. on ) has infinite topological entropy and an invariant measure with full support.
Then weakly mixing sets are introduced and they are characterized, roughly speaking, as the sets on which the transformation acts in a weakly mixing way. The system is called partly mixing when it contains a weakly mixing set. In Theorem 4.5 it is proved that if (resp. ) then the set of those entropy (resp. -entropy) sets which are weakly mixing is a dense subset of (resp. ). The authors show, with the help of natural extensions, that Theorem 4.5 holds also for noninvertible maps. This together with a simple lemma that partial mixing implies Li-Yorke chaos gives a topological proof of the known fact [see the author, E. Glasner, S. Kolyada and A. Maass, J. Reine Angew. Math. 547, 51–68 (2002; Zbl 1059.37006)] that positive entropy implies Li-Yorke chaos. A Devaney chaotic symbolic system which is not partly mixing is constructed.
Two different notions of capacity, one of them being the Bowen entropy , are used in the paper to measure the size of entropy sets. It is shown that when is ergodic with , the set of all weakly mixing -entropy sets such that is residual in . When in addition is uniquely ergodic the set of all weakly mixing entropy sets with is residual in .