Exponential trichotomy and a class of ergodic solutions of differential equations with ergodic perturbations.

*(English)*Zbl 0935.34044The existence of a class of ergodic solutions to some differential equations is investigated by using exponential trichotomy. An application to the Hill equation with an ergodic forcing function is given.

One of the two main theorems is that if $\dot{x}=A\left(t\right)x$ admits an exponential trichotomy then $\dot{x}=A\left(t\right)x+f\left(t\right)$ has at least one ergodic solution for every ergodic $f$, with a special class of ergodic functions. The second theorem allows $f$ to depend on $x$ with a sufficiently small Lipschitz constant.

Reviewer: S.Siegmund (Augsburg)