Nontrivial attractor-repellor maps of $S^2$ and rotation numbers

Abstract

We consider an orientation preserving homeomorphism $h$ of $S^2$ which admits a repellor denoted $\infty$ and an attractor $-\infty$, which is not a North-South map, such that the basins of $\infty$ and $-\infty$ intersect. We study various aspects of the rotation number of $h:S^2\setminus \{\pm \infty \}\to S^2\setminus \{\pm \infty \}$, especially its relationship with the existence of periodic orbits.

arXiv comment24 pages, 6 figure