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 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.