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Difference between revisions of "Harmonizable dynamical system"

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A [[Flow (continuous-time dynamical system)|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).
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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====
 
====Comments====
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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In the above, an almost-periodic trajectory in a [[dynamical system]] $\{ S^t \}$ on a metric space $(X,\rho)$ 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  
 
\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|almost-period]] of an [[Almost-periodic function|almost-periodic function]]).
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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]] of an [[almost-periodic function]]).
  
 
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[[Category:Dynamical systems and ergodic theory]]

Latest revision as of 19:16, 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 $(X,\rho)$ 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=34022
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article