Extensions of dynamical systems

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1987
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Pennings, Timothy
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Justin Peters
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Altmetrics
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Mathematics
Abstract

Let X be compact Hausdorff, [sigma] the natural numbers or integers, [phi]: X → X, and [phi][superscript] k: k ϵ [sigma] a (semi)group of continuous functions from X to X. Given the dynamical system (X,[phi],[sigma]), let U be a [sigma]-invariant C[superscript]*-algebra of bounded functions containing C(X). There is a natural extension (X,[phi],[sigma]) of (X,[phi],[sigma]) where X is the spectrum of U and [phi] is given by [phi](x)f = x(f ∘ [phi]). If U has a dense subset of functions continuous on a cofinite set, then (X, [phi],[sigma]) inherits the properties of minimality and topological transitivity from (X,[phi],[sigma]) if and only if U contains no point characteristic functions;If X is the circle, [phi] an irrational rotation, and U the C[superscript]*-algebra generated by C(X) and [xi] ∘ [phi][superscript] k, k ϵ Z, where the bounded function [xi] is continuous on X 0, then (X,[phi],Z) preserves the zero topological entropy of (X,[phi],Z) but does not inherit the distality of (X,[phi],Z). (X,[phi],Z) may or may not be expansive depending on the function [xi].

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Thu Jan 01 00:00:00 UTC 1987