The geometry of Julia sets

Abstract

Abstract.The long term analysis of dynamical systems inspired the study of the dynamics of families of mappings. Many of these investigations led to the study of the dynamics of mappings on Cantor sets and on intervals. Julia sets play a critical role in the understanding of the dynamics of families of mappings. In this paper we introduce another class of objects (called hairy objects) which share many properties with the Cantor set and the interval: they are topologically unique and admit only one embedding in the plane. These uniqueness properties explain the regular occurrence of hairy objects in pictures of Julia sets-hairy objects are ubiquitous. Hairy arcs will be used to give a complete topological description of the Julia sets of many members of the exponential family.

Trees and hairs for some hyperbolic entire maps of finite order

“…In [11], the existence of so-called Cantor bouquets in the Julia set was proved for a number of families of entire maps from B, including the exponential and sine families from Example 1. In [1], Aarts and Oversteegen showed that for these two families the Julia set is homeomorphic to so-called straight brush in the plane. In particular, each hair is homeomorphic to the half-line [0, ∞).…”

“…Take z ∈ X 1 ∩ X 2 . Then z = lim n→∞ z (1) n = lim n→∞ z (2) n for some z -sup z∈ω X dist(z, X), sup z∈ω Y dist(z, Y) < ε for a small ε > 0, -ω X (0), ω Y (0) ∈ 1 , ω X (1), ω Y (1) ∈ 2 , -Re(ω X (t)), Re(ω Y (t)) ∈ (R 1 , R 2 ) for t ∈ (0, 1),…”

Exotic topology in complex dynamics

“…following Aarts and Oversteegen [1], a Cantor bouquet is any planar set that is homeomorphic to a straight brush. To define this set, let B be a subset of [0, ∞) × I where I is a dense subset of the irrational numbers.…”

“…In the limit, the set of points which do not fall into H after some iterate of E λ is the Julia set of E λ , J (E λ ), and it is known that this set consists of infinitely many curves, each with a distinguished endpoint and a "stem", i.e., the portion of the curve that extends from the endpoint to ∞ in the right half plane. This is the Cantor bouquet [1,9].…”

Do Diophantine vectors form a Cantor bouquet?

“…The problem is to show that R n t;g is a Cantor bouquet [1,5]. For this, it is needed to know more about the endpoints of the closed half lines in R n t;g , in particular, one has to prove that the set of endpoints accumulates on every point of R n t;g .…”