Entropy zero area preserving diffeomorphisms of $S^2$

TL;DR

This paper establishes a structure theorem for zero entropy area-preserving diffeomorphisms of the sphere and explores their relation to mapping class group actions and cohomology.

Contribution

It provides a new structure theorem for such diffeomorphisms and links their existence to properties of mapping class groups and cohomology.

Findings

Structure theorem for zero entropy area-preserving diffeomorphisms of $S^2$

Relation between faithful group actions and cohomology of mapping class groups

Conditions for finite index subgroups with non-trivial first cohomology

Abstract

In this paper we formulate and prove a structure theorem for area preserving diffeomorphisms of genus zero surfaces with zero entropy. As an application we relate the existence of faithful actions of a finite index subgroup of the mapping class group of a closed surface $\Sigma_g$ on $S^2$ by area preserving diffeomorphisms to the existence of finite index subgroups of bounded mapping class groups $MCG(S, \partial S)$ with non-trivial first cohomology.