The General Properties of Discrete-Time Competitive Dynamical Systems

For strongly competitive discrete-time dynamical systems on a strongly ordered topological vector space, it is proved that any α- or ω-limit set is unordered and lies on some invariant hypersurface with codimension one, which generalizes Hirschʹs results for competitive autonomous systems of ordinary differential equations to competitive maps in a very general framework and implies the Sarkovskiiʹs Theorem for planar strongly competitive and monotone maps. We also show that the definition of competitiveness gives more restriction on homeomorphic property.