# Harmonizable dynamical system

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=34024