On groups of diffeomorphisms of the interval with finitely many fixed points I

Azer Akhmedov

Published 2015-03-12Version 1

We strengthen the results of [1], consequently, we improve the claims of [2] 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. By using the concept of semi-archimedean groups, we also show that the classification picture fails in the continuous category.