Abstract:

In this talk, we aim to to prove the following:
THEOREM: Consider a one parameter family of maps ${f_a}_{ainmathbb{D}(a_0,r)}$, with the property that there exists $R>0$, such that for all $ainmathbb{D}(a_0,r)$, $f_a$  has exactly one singular value $s(a)$ in $overline{mathbb{D}(z_0,R)}$ which depends holomorphically on $a$. Suppose also: $f_{a_0}(z_0)=z_0$, $,f_{a_0}'(z_0)=1$, and $,f_{a_0}''(z_0) eq 0$, and the attracting petal of the parabolic fixed point $z_0$ contains only $s(a_0)$ of all singular values (if there are more).

Suppose a forward invariant curve $gamma_{a_0}:[0,infty) ightarrowmathbb{C}$, $tmapsto gamma_{a_0}(t)$
such that $gamma_{a_0}(0)=z_0$ is given. There exists a quasiconformal homeomorphism $h$ from the parameter plane to the dynamical plane for $f_{a_0}$ with the property that $h^{-1}(gamma_{a_0})$ is a simple curve $Gamma$ in $omega$, such that for each point $a=Gamma(t)inomega$: $s(a)=gamma_a(t)$. Moreover,
begin{equation*}
Gamma(t) ightarrow a_0;;mathrm{as};;t ightarrow 0.
end{equation*}