Ping-pong partitions and locally discrete groups of real-analytic circle diffeomorphisms, II: Applications

TL;DR

This paper applies a classification method for certain discrete groups of real-analytic circle diffeomorphisms to solve an old conjecture and to explore examples of minimal actions not of Fuchsian type.

Contribution

It extends the classification framework to derive new results on group actions and provides counterexamples to Fuchsian-type actions.

Findings

Confirmed that actions with invariant Cantor sets are semi-conjugate to piecewise linear actions.

Solved Dippolito's conjecture within this setting.

Constructed examples of minimal actions not of Fuchsian type.

Abstract

In the first part of this work we have established an efficient method to obtain a topological classification of locally discrete, finitely generated, virtually free subgroups of real-analytic circle diffeomorphisms. In this second part we describe several consequences, among which the solution (within this setting) to an old conjecture by P. R. Dippolito [Ann. Math. 107 (1978), 403-453] that actions with invariant Cantor sets must be semi-conjugate to piecewise linear actions. In addition, we exhibit examples of locally discrete, minimal actions which are not of Fuchsian type.