The classical Poincare-Bendixson index theorem expresses an index of a stationary point of a planar vector field (i.e. a continuous dynamical system), by the number of elliptic and hyperbolic regions. During the last decade a counterpart of Poincare-Bendixson index theorem has been proved for planar discrete dynamical systems generated by homeomorphisms (Le Calvez and Yoccoz; Franks, Ruiz del Portal and Salazar). In this talk we discuss these results and sketch a generalization, obtained by the use of Conley index methods, for discrete semi-dynamical systems.
|