Extension of H\"older's Theorem in Diff_{+}^{1+\epsilon}(I)

TL;DR

This paper extends H"older's theorem to certain subgroups of diffeomorphisms with regularity conditions, showing they are solvable or meta-abelian based on fixed point constraints.

Contribution

It generalizes H"older's theorem to subgroups of Diff_{+}^{1+ ext{epsilon}}(I) with fixed point restrictions, establishing solvability and meta-abelian properties.

Findings

Subgroups with bounded fixed points are solvable.

Enhanced regularity (Diff_{+}^{2}(I)) implies meta-abelian structure.

Extension of classical theorems to broader diffeomorphism classes.

Abstract

We prove that if \Gamma is subgroup of Diff_{+}^{1+\epsilon}(I) and N is a natural number such that every non-identity element of \Gamma has at most N fixed points then \Gamma is solvable. If in addition \Gamma is a subgroup of Diff_{+}^{2}(I) then we can claim that \Gamma is metaabelian.