*(English)*Zbl 1151.37019

This is a paper dealing with the local theory of topological entropy for dynamical systems given by a compact metric space $X$ and a homeomorphism $T:X\to X$.

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 $K$ is an entropy set if and only if any tuple of elements of $K$ which are not all identical is an entropy tuple. An analogous result holds for $\mu $-entropy sets where $\mu $ is an invariant measure.

In the paper under review the authors study the families ${E}_{s}(X,T)$ and ${E}_{s}^{\mu}(X,T)$ of entropy sets and $\mu $-entropy sets and their closures $H(X,T)$ and ${H}^{\mu}(X,T)$ in the hyperspace ${2}^{X}$. The sets $H(X,T)\setminus {E}_{s}(X,T)$ and ${H}^{\mu}(X,T)\setminus {E}_{s}^{\mu}(X,T)$ consist only of singletons. Denote by ${h}_{\text{top}}\left(T\right)$ resp. ${h}_{\mu}\left(T\right)$ the topological entropy (resp. measure-theoretical entropy with respect to the measure $\mu $) of $T$. If ${h}_{\text{top}}\left(T\right)>0$ (resp. ${h}_{\mu}\left(T\right)>0$) then the action induced by $T$ on $H(X,T)$ (resp. on ${H}^{\mu}(X,T)$) 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 $(X,T)$ is called partly mixing when it contains a weakly mixing set. In Theorem 4.5 it is proved that if ${h}_{\text{top}}\left(T\right)>0$ (resp. ${h}_{\mu}\left(T\right)>0$) then the set of those entropy (resp. $\mu $-entropy) sets which are weakly mixing is a dense ${G}_{\delta}$ subset of $H(X,T)$ (resp. ${H}^{\mu}(X,T)$). 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 $h$, are used in the paper to measure the size of entropy sets. It is shown that when $\mu $ is ergodic with ${h}_{\mu}\left(T\right)>0$, the set of all weakly mixing $\mu $-entropy sets $E$ such that $h\left(E\right)\ge {h}_{\mu}\left(T\right)$ is residual in ${H}^{\mu}(X,T)$. When in addition $(X,T)$ is uniquely ergodic the set of all weakly mixing entropy sets $E$ with $h\left(E\right)={h}_{\text{top}}\left(T\right)$ is residual in $H(X,T)$.

##### MSC:

37B40 | Topological entropy |

54H20 | Topological dynamics |

28D20 | Entropy and other measure-theoretic invariants |

37A35 | Entropy and other invariants, isomorphism, classification (ergodic theory) |

37B05 | Transformations and group actions with special properties |