Parameter space for families of parabolic-like mappings

In this paper we study families of degree 2 parabolic-like mappings (fλ)λ∈Λ(fλ)λ∈Λ (as defined in [4]). We prove that the hybrid conjugacies between a nice analytic family of degree 2 parabolic-like mappings and members of the family Per1(1)Per1(1) induce a continuous map χ:Λ→Cχ:Λ→C, which under suitable conditions restricts to a ramified covering from the connectedness locus of (fλ)λ∈Λ(fλ)λ∈Λ to the connectedness locus M1∖{1}M1∖{1} of Per1(1)Per1(1). As an application, we prove that the connectedness locus of the family Ca(z)=z+az2+z3Ca(z)=z+az2+z3, a∈Ca∈C presents baby M1M1.