Given a homeomorphism $h$ of the extended real line.
Is it true that there exists an extension $\hat h$ of $h$, which is a Mobius transformation of a hyperbolic space $\mathbb{H}$?
Any hints are welcome! Thank you in advance!
2026-03-28 09:55:54.1774691754
homeomorphisms of the real line
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No. There is an infinite-dimensional space of homeomorphisms of the circle; the space of Mobius transformations is the 3-dimensional $PSL_2(\Bbb R)$. But more pointedly, the restriction of a Mobius transformation to the ideal boundary can have at most two fixed points if it's not the identity; but any closed subset of $S^1$ is the set of fixed points of some homeomorphism.