Paul Blanchard

Sierpinski-curve Julia sets and singular perturbations of complex polynomials

In this paper we consider the family of rational maps of the complex plane given by z 2 + λ z 2 where λ is a complex parameter. We regard this family as a singular perturbation of the simple function z 2 . We show that, in any neighborhood of the origin in the parameter plane, there are infinitely many open sets of parameters for which the Julia sets of the correspondingmaps are Sierpinski curves. Hence all of these Julia sets are homeomorphic. However,we also showthatparameterscorrespondingto differentopensets havedynamics that are not conjugate.

A GENERALIZED VERSION OF THE MCMULLEN DOMAIN

We study the family of complex maps given by Fλ(z) = zn + λ/zn + c where n ≥ 3 is an integer, λ is an arbitrarily small complex parameter, and c is chosen to be the center of a hyperbolic component of the corresponding Multibrot set. We focus on the structure of the Julia set for a map of this form generalizing a result of McMullen. We prove that it consists of a countable collection of Cantor sets of closed curves and an uncountable number of point components.

DISCONNECTED JULIA SETS

Publisher Summary This chapter discusses disconnected Julia sets. The connectivity properties of the Julia set for a polynomial have an intimate relationship with the dynamical properties of the finite critical points. The chapter explains the way to construct symbolic codings for the components of the Julia set for a large class of cubic polynomials. These cubics can have one critical point that iterates to infinity and another whose orbit remains bounded. The chapter discusses two filled-in Julia sets for two different types of polynomials. The black regions are the filled-in Julia set and the shading of the stable manifold of infinity corresponds to levels of the rate of escape map. In the dynamics of high-degree polynomials and in transcendental functions, there are regions on which the dynamics is determined by a low-degree polynomial.

CHECKERBOARD JULIA SETS FOR RATIONAL MAPS

In this paper, we consider the family of rational maps where n ≥ 2, d ≥ 1, and λ ∈ ℂ. We consider the case where λ lies in the main cardioid of one of the n - 1 principal Mandelbrot sets in these families. We show that the Julia sets of these maps are always homeomorphic. However, two such maps Fλ and Fμ are conjugate on these Julia sets only if the parameters at the centers of the given cardioids satisfy μ = νj(d+1)λ or where j ∈ ℤ and ν is an (n - 1)th root of unity. We define a dynamical invariant, which we call the minimal rotation number. It determines which of these maps are conjugate on their Julia sets, and we obtain an exact count of the number of distinct conjugacy classes of maps drawn from these main cardioids.

Complex dynamics: an informal discussion

This paper is an informal introduction to the theory of complex-analytic dynamical systems. We survey the general theory, and then discuss two important classes of examples — quadratics and Newton’s method. We also present the Douady-Hubbard theory of polynomial-like mappings.