Differentiable conjugacy for groups of area preserving circle diffeomorphisms

Convergence groups and semiconjugacy

“…If γ is conjugate to δ 1 , then Proposition 4.1, states that there is a continuous volume form ω γ on M h that is invariant under (ρ 1 (γ), ρ 2 (γ)). If γ is conjugate to δ 2 , then Proposition 1.7 of [Mon14a] gives the same result (the proof is almost identical to Proposition 4.1).…”

“…Proposition 4.1 shows that the images of the generators by (ρ 1 , ρ 2 ) each preserve a volume form, and the difficulty consists in showing that we can find one that is preserved by all three. The proof is almost identical to the proof of Theorem 1.4 in [Mon14a].…”

“…We only showed that ω is continuous, but following [Mon14a] one can show that it is possible to obtain smooth volume form.…”

“…In [Mon14a], we studied the case where h = Id (hence ρ 1 = ρ 2 ). The main example is given by the projective action of PSL(2, R) on S 1 ≈ RP 1 and the volume form 4 (x−y) 2 dx ∧ dy.…”

Isometries of Lorentz surfaces and convergence groups

“…Such a classification would require a characterisation of ρ M 1 , ρ M 2 up to conjugacy in Diff(S 1 ). This problem is addressed [Mon14a] in the case where (M, g) is conformal to the De Sitter space dS 2 . 1.4.5.…”