Multivalued maps, selections and dynamical systems

Under suitable hypotheses the well known notion of first prolongational set J + gives rise to a multivalued map ψ : X → 2 X which is continuous when the upper semifinite topology is considered in the hyperspace of X. Some important dynamical concepts such as stability or attraction can be easily characterized in terms of ψ and moreover, the classical result that an attractor in R n has the shape of a finite polyhedron can be reinforced under the hypotheses that the mapping ψ is small and has a selection.