# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Entropy sets, weakly mixing sets and entropy capacity. (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 ${E}_{s}\left(X,T\right)$ and ${E}_{s}^{\mu }\left(X,T\right)$ of entropy sets and $\mu$-entropy sets and their closures $H\left(X,T\right)$ and ${H}^{\mu }\left(X,T\right)$ in the hyperspace ${2}^{X}$. The sets $H\left(X,T\right)\setminus {E}_{s}\left(X,T\right)$ and ${H}^{\mu }\left(X,T\right)\setminus {E}_{s}^{\mu }\left(X,T\right)$ 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 $H\left(X,T\right)$ (resp. on ${H}^{\mu }\left(X,T\right)$) 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 $\left(X,T\right)$ 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 $H\left(X,T\right)$ (resp. ${H}^{\mu }\left(X,T\right)$). 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 ${H}^{\mu }\left(X,T\right)$. When in addition $\left(X,T\right)$ 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 $H\left(X,T\right)$.

##### 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