Abstract:

Let f = (fλ : U λ → U λ ) be an analytic family of parabolic-like maps of
degree 2 parametrized by an open set Λ, which is isomorphic to the unit
disk. Let Mf = {λ | Kλ is connected } be its connectedness locus. Every
fλ is hybrid equivalent to a member of the family of quadratic rational
maps of the form PA(z ) = z + 1/z + √A, z ∈ C. If λ ∈ Mf this rational
map PA is unique, then we can define a map
χ : Mf → C
λ → √A.
We will prove that, if the boundaries of Uλ move holomorphically and Mf
is compact, then the map χ extends to
χ : Λ → C,
which is homeomorphism between Mf and M1 , and quasiregular on Λ M f .