On groups of homeomorphisms of the interval with finitely many 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 Hölder'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.

arXiv commentWe have strengthened the results of the previous version of the paper