If $F=(f_\lambda)_{\lambda\in\Lambda}$, is an analytic family of quadratic-like maps (with some conditions), and $M_F$ is the set of $f_\lambda$ with connected Julia set, then there is a homeomorphism of $M_F$ with the Mandelbrot set $\{c\in\mathbb{C}:f_c(z)=z^2+c\text{ has connected Julia set}\}$. See http://www.math.cornell.edu/~hubbard/PolyLikeMaps.pdf. Douady called this the "miracle of continuity" coming from the measurable Riemann mapping theorem.

Analytic families of polynomial-like maps of degree $\geq 3$ have discontinuous straighting maps. See http://arxiv.org/pdf/0903.4289v2.pdf.