Abstract:

In the talk I will give an example of an application of holomorphic motion. We will study a family of quadratic rational maps, with fixed points at infinity and the origin. The family is parametrized by the two multipliers at these fixed points. We will restrict attention to the cases where the fixed point at infinity is attracting, i.e. the multiplier $\lambda$ at infinity is in the unit disk. We will consider slices of the parameter space, for fixed $\lambda$ in $D$, or more precisely we will consider the connectedness loci M_$\lambda$ in these slices. Then holomorphic motion is used as a tool to show that for all $\lambda$, $\lambda \prime$ in $D$, the connectedness loci $M$_λ and $M$_λ′ are quasiconformally equivalent.