Boundedness of hyperbolic components in moduli space.

ABSTRACT: A complex rational map of degree at least 2 is hyperbolic if each of its critical points is attracted to an attracting cycle. For a fixed degree, the hyperbolic rational maps form an open set in the space of rational maps. This open set deduces an open set in the moduli space of rational maps, modulo the Möbius conjugacy. Each component of the deduced open set is a hyperbolic component. In this talk, I will present some precompactness results on hyperbolic components. In particular, I will focus on the space of quartic Newton maps. This is a joint work with Y. Gao.