Density of hyperbolicity in families of complex rational maps

Abstract

In this work, we address the fundamental open problem of whether hyperbolic rational maps, the ones for which every critical point lies in the basin of an attracting cycle, are dense in the space of rational maps of the same degree. By this, we mean if any such map can be uniformly approximated on compact sets by hyperbolic ones. Conjecturally, the answer is ’yes’, and this is known as the Density of Hyperbolicity Conjecture. After reviewing key tools from complex dynamics such as puzzle pieces constructions, quasi-conformal conjugacies, Böttcher coordinates and holomorphic motions, we introduce complex box mappings as a natural extension of polynomiallike maps and discuss their rigidity under combinatorial equivalence. Focusing on non-renormalisable polynomials without neutral periodic points, we reproduce, clarify and check the Kozlovski–van Strien result that such polynomials admit approximating hyperbolic maps by constructing dynamically natural box mappings and applying topological and rigidity results. In conclusion, we outline how this framework, with a careful setting, promises to extend beyond the polynomial case to prove density of hyperbolicity in broader families such as Newton and McMullen maps, thereby sketching a clear path for future advances in complex dynamics.