Dynamics of the mapping class group on the moduli of a punctured sphere with rational holonomy

Let M be a four-holed sphere and Γ the mapping class group of M fixing the boundary ∂M. The group Γ acts on M _B(SL(2,ℂ)) = Hom_B⁺(π₁(M), SL(2,ℂ))/SL(2,ℂ) which is the space of completely reducible SL(2,ℂ)-gauge equivalence classes of flat SL(2,ℂ)-connections on M with fixed holonomy B on ∂M. Let B∈[-2,2]⁴ and M _B be the compact component of the real points of M _B(SL(2,ℂ)). These points correspond to SU(2)-representations or SL(2,ℝ)-representations. The Γ-action preserves M_B and we study the topological dynamics of the Γ-action on M_B and show that for a dense set of holonomy B∈[-2, 2]⁴, the Γ-orbits are dense in M_B. We also produce a class of representations ρ∈Hom_B⁺(π₁(M), SL(2,ℝ)) such that the Γ-orbit of [ρ] is finite in the compact component of M _B(SL(2,ℝ)), but ρ(π₁(M)) is dense in SL(2, ℝ}.