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

TL;DR

This paper investigates orientation-preserving homeomorphisms of the 2-sphere with intersecting attractor and repellor basins, analyzing their rotation numbers and the connection to periodic orbits, revealing complex dynamical behaviors.

Contribution

It introduces a detailed study of rotation numbers for nontrivial attractor-repellor maps on the sphere, extending understanding beyond simple North-South dynamics.

Findings

Rotation numbers relate to the existence of periodic orbits.

Intersections of basins lead to complex rotation behaviors.

The study characterizes dynamics of nontrivial attractor-repellor maps.

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.