Difference between revisions of "Harmonizable dynamical system"
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| − | A [[ | + | 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 [[ | + | 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 [[ | + | 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).
Harmonizable dynamical system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonizable_dynamical_system&oldid=34022