Global Behavior of Four Competitive Rational Systems of Difference Equations in the Plane

We investigate the global dynamics of solutions of four distinct competitive rational systems of difference equations in the plane. We show that the basins of attractions of different locally asymptotically stable equilibrium points are separated by the global stable manifolds of either saddle points or nonhyperbolic equilibrium points. Our results give complete answer to Open Problem 2 posed recently by Camouzis et al. (2009).

Fatou maps inℙndynamics

We study the dynamics of a holomorphic self-mapfof complex projective space of degreed>1by utilizing the notion of a Fatou map, introduced originally by Ueda (1997) and independently by the author (2000). A Fatou map is intuitively like an analytic subvariety on which the dynamics offare a normal family (such as a local stable manifold of a hyperbolic periodic point). We show that global stable manifolds of hyperbolic fixed points are given by Fatou maps. We further show that they are necessarily Kobayashi hyperbolic and are always ramified byf(and therefore any hyperbolic periodic point attracts a point of the critical set off). We also show that Fatou components are hyperbolically embedded inℙnand that a Fatou component which is attracted to a taut subset of itself is necessarily taut.