The topological dynamics of semigroup actions (original) (raw)

Compactifications of Semigroups and Semigroup Actions

2006

An action of a topological semigroup S on X is compactifiable if this action is a restriction of a jointly continuous action of S on a Hausdorff compact space Y. A topological semigroup S is compactifiable if the left action of S on itself is compactifiable. It is well known that every Hausdorff topological group is compactifiable. This result cannot

Recurrent sets and shadowing for finitely generated semigroup actions on metric spaces

Hacettepe Journal of Mathematics and Statistics, 2021

We introduce various new type of recurrent sets for finitely generated semigroups on noncompact metric spaces that are conjugacy invariant, and obtain some basic properties of chain recurrent sets for semigroups via these new definitions. Moreover, we define the notion of weak shadowing property for finitely generated group actions on compact metric spaces, which is weaker than that of shadowing property, and prove the equivalence of the shadowing and weak shadowing properties for the finitely generated group actions on a generalized homogeneous space without isolated points.

Recurrence in generalized semigroup

Indian Journal of Pure and Applied Mathematics

In [3], we introduced the concept of escaping set in general setting for a topological space and extended the notion of limit set and escaping set for the general semigroup generated by continuous self maps. In this paper we continue with extending the other notions of recurrence for the generalized semigroup analogs to their counterpart in the classical theory of dynamics. We discuss the concept of periodic point, nonwandering point and chain recurrent point in the more general setting of a semigroup and establish the correlation between them.

Noncompact semigroup actions

Semigroup Forum, 1974

Communicated by K. H. Hofmann I. INTRODUCTION This paper attempts to characterize certain noncompact semigroup actions by means of noncompact transformation group methods. The semigroup actions involved are those of locally compact groups with compact boundaries, that is, semigroups which are the union of two disjoint groups, one compact and one noncompact with the compact group contained in the closure of the noncompact group. Hofmann [5] has characterized these semigroups and we rely strongly on his results. The study of actions of a locally compact group with zero, a special case of actions of a locally compact group with compact boundary, began with Hanson [3, 4], using cross-sections and then King [8] using slices, a tool well-known in the study of transformation groups. In this paper we decompose the action of a locally compact group with compact boundary T = HuB on a topological space X into two transformation groups, (X-eX,H) which is noncompact and (eX,B) which is compact where e is the identity of B. We characterize each in terms of slices. We also present some results involving the interaction of B and H on X although much more along these lines needs to be done.

Semi-Asymptotic Non-Expansive Actions of Semi-Topological Semigroups

Bulletin of the Korean Mathematical Society, 2016

In this paper we extend Takahashi's fixed point theorem on discrete semigroups to general semi-topological semigroups. Next we define the semi-asymptotic non-expansive action of semi-topological semigroups to give a partial affirmative answer to an open problem raised by A.T-M. Lau.

Banach Representations and Affine Compactifications of Dynamical Systems

Fields Institute Communications, 2013

To every Banach space V we associate a compact right topological affine semigroup E(V ). We show that a separable Banach space V is Asplund if and only if E(V ) is metrizable, and it is Rosenthal (i.e. it does not contain an isomorphic copy of l 1 ) if and only if E(V ) is a Rosenthal compactum. We study representations of compact right topological semigroups in E(V ). In particular, representations of tame and HNS-semigroups arise naturally as enveloping semigroups of tame and HNS (hereditarily non-sensitive) dynamical systems, respectively. As an application we obtain a generalization of a theorem of R. Ellis. A main theme of our investigation is the relationship between the enveloping semigroup of a dynamical system X and the enveloping semigroup of its various affine compactifications Q(X). When the two coincide we say that the affine compactification Q(X) is E-compatible. This is a refinement of the notion of injectivity. We show that distal non-equicontinuous systems do not admit any E-compatible compactification. We present several new examples of non-injective dynamical systems and examine the relationship between injectivity and E-compatibility.

Reductive compactifications of semitopological semigroups

International Journal of Mathematics and Mathematical Sciences

We consider the enveloping semigroup of a flow generated by the action of a semitopological semigroup on any of its semigroup compactifications and explore the possibility of its being one of the known semigroup compactifications again. In this way, we introduce the notion of E-algebra, and show that this notion is closely related to the reductivity of the semigroup compactification involved. Moreover, the structure of the universal Eℱ-compactification is also given.

A note on sensitivity of semigroup actions

Semigroup Forum, 2008

It is well known that for a transitive dynamical system (X, f ) sensitivity to initial conditions follows from the assumption that the periodic points are dense. This was done by several authors: Banks, Brooks, Cairns, Davis and Stacey [2], Silverman and Glasner and Weiss . In the latter article Glasner and Weiss established a stronger result (for compact metric systems) which implies that a transitive non-minimal compact metric system (X, f ) with dense set of almost periodic points is sensitive. This is true also for group actions as was proved in the book of Glasner .

Attractors and chain recurrence in generalized semigroup

arXiv: Dynamical Systems, 2019

In \cite {kl1}, we extended various notions of recurrence for the generalized semigroup analogous to their counterpart in the classical theory of dynamics. In this paper, we shall address the alternative definition of chain recurrent set in terms of attractors, given by Hurley in \cite {mh} following Conley\rq{}s characterization in \cite {conley}. We shall also discuss the notion of topological transitivity and chain transitivity in this general setting.