Rotation Numbers for $S^2$ diffeomorphisms

TL;DR

This paper explores the properties of a function that assigns a numerical value to quadruples of fixed points in orientation-preserving homeomorphisms or diffeomorphisms of the 2-sphere, providing insights into their rotational behavior.

Contribution

It offers an in-depth, largely expository analysis of the function ${ m extbf{ extit R}}$ related to fixed points of $S^2$ diffeomorphisms, clarifying its properties and significance.

Findings

Characterization of the function ${ m extbf{ extit R}}$ for fixed points

Properties of ${ m extbf{ extit R}}$ under different conditions

Implications for understanding $S^2$ diffeomorphisms

Abstract

These largely expository notes describe the properties of the function ${\cal R}$ which assigns a number to a $4$-tuple of distinct fixed points of an orientation preserving homeomorphism or diffeomorphism of $S^2$.