On the use of the topological degree theory in broken orbits analysis

Dynamical systems $f$ in ${\mathbb R}^{d}$ are studied. Let $\mbox{\boldmath $\Omega$ } \subset{\mathbb R}^{d}$ be a bounded open set. We will be interested in those periodic orbits such that at least one of its points lies inside $\mbox{\boldmath$\Omega$ }$ and at least one of its points lies outside $\overline{\mbox{\boldmath$\Omega$ }}$; the orbits with this property are called $\mbox{\boldmath$\Omega$ }$-broken. Information about the structure of the set of $\mbox{\boldmath$\Omega$ }$-broken orbits is suggested; results are formulated in terms of topological degree theory.