Cubic Polynomial Maps with Periodic Critical Orbit, Part I

Abstract

This paper concerns parameter spaces of rational maps viewed as dynamical systems through iteration. The parameter space of a family of rational maps often has a natural decomposition into the hyperbolic locus and the non-hyperbolic locus: a rational map is hyperbolic if and only if all its critical points are attracted to attracting periodic orbits. The hyperbolic locus is open (and conjecturally dense, in any reasonable family). A hyperbolic component is a connected component of the hyperbolic locus. Maps within the same hyperbolic component have essentially the same macroscopic dynamics.