One hundred years of complex dynamics

Abstract

Holomorphic dynamics has been a key area of study for 100 years. In this survey, we focus on the dynamics of rational functions.

Correspondences in Complex Dynamics

“…A map f a is structurally stable if f c is topologically conjugate to f a , for every c in an open set containing a. For rational maps on the Riemann sphere J-stability, which roughly speaking means stability on a neighborhood of the Julia set, is usually considered [4]. Mañé, Sad and Sullivan [7] have shown that the set of J-structurally stable rational maps is open and dense in the space of rational maps Rat d of the same degree.…”

“…In 1994 the first author and C. Penrose proved that for all parameters a in the real interval [4,7], the correspondence F a is a mating between a quadratic polynomial f c (z) = z 2 + c, c ∈ [−2, +1/4] ⊂ R and the modular group Γ = P SL(2, Z) (see [1]).…”

“…Consider the holomorphic correspondence H on the upper half-plane obtained from the generators α(z) = z +1 and β(z) = z/(z +1) of PSL(2, Z), i.e. defined by the polynomial equation (4). As part of the proof of Theorem 2.1 it is shown in [16] that: Theorem 2.2 (Böttcher map) If a ∈ C Γ , there is a unique conformal homemorphism ϕ a : Ω a → H such that…”

“…Theorem 1.1 For every a in the real interval [4,7], the correspondence F a is a mating between some quadratic map f c (z) = z 2 + c and the modular group Γ = PSL(2, Z), and conjectured that the connectedness locus for this family is homeomorphic to the Mandelbrot set.…”

“…Excellent sources for details are the books of Milnor [2] and de Faria and de Melo [3]. An overview of a century of complex dynamics is presented in the article by Mary Rees [4].…”