On proper extensions of the conformal group of sphere diffeomorphisms

TL;DR

This paper characterizes the structure of groups of sphere diffeomorphisms extending conformal transformations, showing they must be highly transitive and contain elements with complex dynamics.

Contribution

It proves that such extension groups are at least 4-transitive and contain positive entropy elements, providing a new characterization of Möbius transformations.

Findings

Extension groups are at least 4-transitive.

Such groups always contain elements with positive topological entropy.

A characterization of Möbius transformations via transitivity.

Abstract

In this paper, we prove that any group of diffeomorphisms acting on the 2-sphere and properly extending the conformal group of M\"obius transformations must be at least 4-transitive or, more precisely, arc 4-transitive. In addition, we show that any such group must always contain an element of positive topological entropy. We also provide an elementary characterization, in terms of transitivity, of the M\"obius transformations within the full group of sphere diffeomorphisms.