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Title: Dynamical Systems from Uniform Completions
Authors: Garibay, F.
Sanchis, M.
Vera, R.
Keywords: uniform completioncompactificationdynamical systemminimalitytransitivity
Issue Date: 2001
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.
Series/Report no.: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
32 (2001) suppl.2
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.
Type: Article
ISSN: 0049-4704
Appears in Collections:Rendiconti dell'Istituto di Matematica dell'Università di Trieste: an International Journal of Mathematics vol.32 (2001) s2

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