Harmonizable dynamical system

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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).


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:
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article