Modulus of Continuity for Finite Energy Homeomorphisms of the Circle

Given $h_o: \mathbb{S} \to \mathbb{S}$, an orientation-preserving homeomorphism of the unit circle, we seek to establish very weak conditions so as to derive a modulus of continuity for $h_o$ and its inverse $h_o^{-1}$, then use these to derive various compactness properties for families of circle h