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Dynamical Systems from Uniform Completions
Garibay, F.
Sanchis, M.
Vera, R.
2001
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.
Series
Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
32 (2001) suppl.2
Publisher
Università degli Studi di Trieste. Dipartimento di Scienze Matematiche
Source
F. Garibay, M. Sanchis and R. Vera, "Dynamical Systems from Uniform Completions", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 32 (2001) suppl.2, pp. 47–57.
Languages
en
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