Dynamical Systems from Uniform Completions

Abstract

Let $\left(X,\mathcal{U}\right)$ be a compact uniform space, $\sum$ the set of natural numbers or the integers, $\varphi\;:\; X\;\longrightarrow\; X$ a continuous function or a homeomorphism. Given the dynamical system $\left(X,\varphi,\sum\right)$, an extension $\left(K,\widehat{\varphi,}\sum\right)$, can be constructed by letting K be the uniform completion of $\left(X,\mathcal{V}\right)$, where $\mathcal{V}$ is a totally bounded uniformity fi{}ner than $\mathcal{U}$. If D$_{f}$ means for the set \[ \left\{ x\:\epsilon\: X\:\mid\: f\::(X,\mathcal{U})\longrightarrow\mathbb{C}\; is\; discontinuous\; at\; x\right\} , \] we prove that, if C(K) contains a dense subset E which contains no characteristic functions of singletons and such that, for each $f\epsilon E$ , there exists a fi{}nite subset F of D$_{f}$ with $D_{f}\backslash F$ discrete (in $\left(X,\mathcal{U}\right)$), then $\left(K,\widehat{\varphi,}\sum\right)$ inherits the properties of minimality and topological transitivity from $\left(X,\varphi,\sum\right)$. Several open questions are posed.