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Difference between revisions of "Harmonizable dynamical system"

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====Comments====
 
====Comments====
In the above, an almost-periodic trajectory in a [[Dynamical system|dynamical system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046580/h0465801.png" /> on a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046580/h0465802.png" /> is the trajectory of any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046580/h0465803.png" /> which has the following property: For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046580/h0465804.png" /> the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046580/h0465806.png" />-almost periods,
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In the above, an almost-periodic trajectory in a [[Dynamical system|dynamical system]] $\{ S^t \}$ on a metric space $X$ is the trajectory of any point $x \in X$ which has the following property: For every $\epsilon > 0$ the set of $\epsilon$-almost periods,
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$$
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\left\lbrace{ \tau \in \mathbb{R} : \rho(S^t(x), S^{t+\tau}(x)) < \epsilon \ \text{for}\ -\infty < t < +\infty }\right\rbrace
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046580/h0465807.png" /></td> </tr></table>
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is relatively-dense in $\mathbb{R}$, that is, there exists an $l(\epsilon) > 0$ such that every interval of length $l(\epsilon)$ contains an $\epsilon$-almost period (compare this with the definition of [[Almost-period|almost-period]] of an [[Almost-periodic function|almost-periodic function]]).
  
is relatively-dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046580/h0465808.png" />, that is, there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046580/h0465809.png" /> such that every interval of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046580/h04658010.png" /> contains an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046580/h04658011.png" />-almost period (compare this with the definition of [[Almost-period|almost-period]] of an [[Almost-periodic function|almost-periodic function]]).
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Revision as of 19:14, 25 October 2014

A flow (continuous-time dynamical system) the trajectories of which become almost-periodic after a certain change of the time. An additional condition which is usually made is that each trajectory be everywhere-dense in the phase space (so that one may speak of a harmonizable minimal set).


Comments

In the above, an almost-periodic trajectory in a dynamical system $\{ S^t \}$ on a metric space $X$ is the trajectory of any point $x \in X$ which has the following property: For every $\epsilon > 0$ the set of $\epsilon$-almost periods, $$ \left\lbrace{ \tau \in \mathbb{R} : \rho(S^t(x), S^{t+\tau}(x)) < \epsilon \ \text{for}\ -\infty < t < +\infty }\right\rbrace $$

is relatively-dense in $\mathbb{R}$, that is, there exists an $l(\epsilon) > 0$ such that every interval of length $l(\epsilon)$ contains an $\epsilon$-almost period (compare this with the definition of almost-period of an almost-periodic function).

How to Cite This Entry:
Harmonizable dynamical system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonizable_dynamical_system&oldid=12736
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article