Locally equicontinuous dynamical systems.

*(English)*Zbl 0976.54044The authors introduce a new class of dynamical systems called locally equicontinuous systems. Let \(X\) be a compact Hausdorff space and \(T\) a self-homeomorphism of it. Assume that \(X\) is metrizable and equipped with a metric \(d\). A point \(x_0\in X\) is said to be an equicontinuity point if for every \(\varepsilon> 0\) there is \(\delta> 0\) such that \(d(x,x_0)< \delta\) implies \(d(T^nx, T^nx_0)< \varepsilon\), for all \(n\in \mathbb{Z}\). The dynamical system on \(X\) induced by \(T\) is locally equicontinuous if each point \(x\in X\) is a point of equicontinuity of the subsystem \(Y=\{\)closure of the orbit of \(x\}\subseteq X\). Denote by WAP, LE, and AE the classes of dynamical systems: weakly almost periodic, locally equicontinuous and almost equicontinuous, respectively. The authors prove that \(\text{WAP} \subset \text{LE} \subset \text{AE}\) and both the inclusions are proper. The main result of the paper is to produce a family of examples of LE dynamical systems which are not WAP. The authors investigate also some properties of LE systems expressed by means of ergodic theory. The following theorem is proved:

Theorem 1.3. Let \((X,T)\) be an LE dynamical system. Then: (1) Every minimal subsystem of \((X,T)\) is equicontinuous, hence it is isomorphic to a group rotation. (2) Every invariant ergodic probability measure on \(X\) is supported on a minimal subsystem and is therefore isomorphic to Haar measure on a group rotation. Every invariant probability measure on \(X\) is supported by the union of the minimal subsystems of \(X\); in particular, if \(X\) has a unique minimal subset then \((X,T)\) is uniquely ergodic.

Theorem 1.3. Let \((X,T)\) be an LE dynamical system. Then: (1) Every minimal subsystem of \((X,T)\) is equicontinuous, hence it is isomorphic to a group rotation. (2) Every invariant ergodic probability measure on \(X\) is supported on a minimal subsystem and is therefore isomorphic to Haar measure on a group rotation. Every invariant probability measure on \(X\) is supported by the union of the minimal subsystems of \(X\); in particular, if \(X\) has a unique minimal subset then \((X,T)\) is uniquely ergodic.

Reviewer: Andrzej Pelczar (Kraków)

##### MSC:

54H20 | Topological dynamics (MSC2010) |

37C55 | Periodic and quasi-periodic flows and diffeomorphisms |

37A99 | Ergodic theory |