Nontrivial attractor-repellor maps of S² and rotation numbers

Abstract

We consider an orientation preserving homeomorphism h of S² which admits a repellor denoted ∞ and an attractor −∞ such that h is not a North-South map and that the basins of ∞ and −∞ intersect. We study various aspects of the rotation number of h: S²\{±∞} → S²\{±∞}, especially its relationship with the existence of periodic orbits.