Related papers: On the Ellis semigroup of a cascade on a compact m…

Given a dynamical system $(X,f)$, we let $E(X,f)$ denote its Ellis semigroup and $E(X,f)^* = E(X,f) \setminus \{f^n : n \in \mathbb{N}\}$. We analyze the Ellis semigroup of a dynamical system having a compact metric countable space as a…

We consider discrete dynamical systems whose phase spaces are compact metrizable countable spaces. In the first part of the article, we study some properties that guarantee the continuity of all functions of the corresponding Ellis…

Let $E(X,f)$ be the Ellis semigroup of a dynamical system $(X,f)$ where $X$ is a compact metric space. We analyze the cardinality of $E(X,f)$ for a compact countable metric space $X$. A characterization when $E(X,f)$ and $E(X,f)^* = E(X,f)…

We introduce the {\it Ellis semigroup} of a nonautonomous discrete dynamical system $(X,f_{1,\infty})$ when $X$ is a metric compact space. The underlying set of this semigroup is the pointwise closure of $\{f\sp{n}_1 \, |\, n\in…

Let $\pi\colon T\times X\rightarrow X$ with phase map $(t,x)\mapsto tx$, denoted $(\pi,T,X)$, be a \textit{semiflow} on a compact Hausdorff space $X$ with phase semigroup $T$. If each $t\in T$ is onto, $(\pi,T,X)$ is called surjective; and…

By proving that, if the quotient space S(X) of the connected components of the locally compact metric space (X,d) is compact, then the full group I(X,d) of isometries of X is closed in C(X,X) with respect to the pointwise topology, i.e.,…

The Ellis semigroup of a dynamical system $(X,T)$ is tame if every element is the limit of a sequence (as opposed to a net) of homeomorphisms coming from the $T$ action. This topological property is related to the cardinality of the…

We show that a homeomorphism of a semi-locally connected compact metric space is equicontinuous if and only if the distance between the iterates of a given point and a given subcontinuum (not containing that point) is bounded away from…

The notion of a proper Ellis semigroup compactification is introduced. Ellis's functional approach shows how to obtain them from totally bounded equiuniformities on a phase space $X$ when the acting group $G$ is with the topology of…

Let $X$ be a compact metric space and let $f:X\rightarrow X$ be a homeomorphism on $X$. We show that if $f$ is both pointwise recurrent and expansive, then the dynamical system $(X, f)$ is topologically conjugate to a subshift of some…

Ellis's "functional approach" allows one to obtain proper compactifications of a topological group $G$ if $G$ can be represented as a subgroup of the homeomorphism group of a space $X$ in the topology of pointwise convergence and $G$-space…

When a topological group $G$ acts on a compact space $X$, its enveloping semigroup $E(X)$ is the closure of the set of $g$-translations, $g\in G$, in the compact space $X^X$. Assume that $X$ is metrizable. It has recently been shown by the…

For topological dynamical systems $(X,T,\sigma)$ with abelian group $T$ which admit an equicontinuous factor $\pi:(X,T,\sigma)\to (Y,T,\delta)$ the Ellis semigroup $E(X)$ is an extension of $Y$ by its subsemigroup $E^{fib}(X)$ of elements…

In this article, we relate the dynamics of a flow $(X, T)$ with the dynamics of the induced flow $(E(X), T)$ where $E(X)$ is the enveloping semigroup of flow $(X, T)$. We establish that a flow $(X, T)$ is distal if and only if the induced…

Let $X$ be a zero-dimensional locally compact Hausdorff space not necessarily metric and $G$ a compactly generated topological group not necessarily abelian or countable. We define recurrence at a point for any continuous action of $G$ on…

Let $X$ be a locally compact zero-dimensional space, let $S$ be an equicontinuous set of homeomorphisms such that $1 \in S = S^{-1}$, and suppose that $\overline{Gx}$ is compact for each $x \in X$, where $G = \langle S \rangle$. We show in…

For an infinite discrete group $G$ acting on a compact metric space $X$, we introduce several weak versions of equicontinuity along subsets of $G$ and show that if a minimal system $(X,G)$ admits an invariant measure then $(X,G)$ is distal…

It is shown that the Ellis semigroup of a $\mathbb Z$-action on a compact totally disconnected space is completely regular if and only if forward proximality coincides with forward asymptoticity and backward proximality coincides with…

We consider locally equi-continuous strongly continuous semigroups on locally convex spaces (X,tau). First, we show that if (X,tau) has the property that weak* compact sets of the dual are equi-continuous, then strong continuity of the…

If $X$ is a finite tree and $f \colon X \longrightarrow X$ is a map, as the Main Theorem of this paper we find eight conditions, each of which is equivalent to the fact that $f$ is equicontinuous. To name just a few of the results obtained:…