Möbius-like groups of homeomorphisms of the circle

Citation

Kovačević, Nataša (1994) Möbius-like groups of homeomorphisms of the circle. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/vy4w-j074. https://resolver.caltech.edu/CaltechTHESIS:05082013-102143473

Abstract

A group G → Homeo_+(S^1) is a Möbius-like group if every element of G is conjugate in Homeo(S^1) to a Mobius transformation. Our main result is: given a Mobus like like group G which has at least one global fixed point, G is conjugate in Homeo(S^1) to a Möbius group if and only if the limit set of G is all of S^1 . Moreover, we prove that if the limit set of G is not SI, then after identifying some closed subintervals of S^1 to points, the induced action of G is conjugate to an action of a Möbius group.

We also show that the above result does not hold in the case when G has no global fixed points. Namely, we construct examples of Möbius-like groups with limit set equal to S^1, but these groups cannot be conjugated to Möbius groups.

Thesis Files