On groups of homeomorphisms of the interval with finitely many fixed points

TL;DR

This paper improves previous results on groups of homeomorphisms of the interval, showing non-discreteness under certain conditions and extending classification theorems for subgroups with limited fixed points.

Contribution

It strengthens existing theorems by proving non-discreteness for subgroups containing free semigroups and extends classification results to broader classes of homeomorphisms.

Findings

Subgroups with free semigroups are not $C_0$-discrete.

Extended H"older's Theorem for subgroups with limited fixed points.

Classification of subgroups where each non-identity element has at most N fixed points.

Abstract

We strengthen the results of \cite{A1}, consequently, we improve the claims of \cite{A2} obtaining the best possible results. Namely, we prove that if a subgroup $\Gamma $ of $\mathrm{Diff}_{+}(I)$ contains a free semigroup on two generators then $\Gamma $ is not $C_0$-discrete. Using this, we extend the H\"older's Theorem in $\mathrm{Diff}_{+}(I)$ classifying all subgroups where every non-identity element has at most $N$ fixed points. In addition, we obtain a non-discreteness result in a subclass of homeomorghisms which allows to extend the classification result to all subgroups of $\mathrm{Homeo}_{+}(I)$ where every non-identity element has at most $N$ fixed points.