Sur les systèmes dynamiques instables

Abstract

Let (X, Φ) be a continuous dynamical system on a locally compact space X with countable base. In this note we prove the equivalence of the following statements:

• 1. (X, Φ) is unstable;

• 2. The kernel $f \mapsto Vf = \int {f(\Phi (t, \cdot ))dt}$, is a proper kernel.

As application, every unstable dynamical system possesses a section S in the form S = {p = q}, such that p and q are lower semicontinuous and >0 on X.