Attractors with irrational rotation number

Abstract

Let h : R-2 -> R-2 be a dissipative and orientation preserving homeomorphism having an asymptotically stable fixed point. Let U be the region of attraction and assume that it is proper and unbounded. Using Caratheodory's prime ends theory one can associate a rotation number, rho, to h(vertical bar U). We prove that any map in the above conditions and with rho is not an element of Q induces a Denjoy homeomorphism in the circle of prime ends. We also present some explicit examples of maps in this class and we show that, if the infinity point is accessible by an arc in U, rho is not an element of Q if and only if Per(h) = Fix(h) = {p}.