The geometry of Julia sets

Authors:
Jan M. Aarts and Lex G. Oversteegen

Journal:
Trans. Amer. Math. Soc. **338** (1993), 897-918

MSC:
Primary 30D05; Secondary 54F65, 54H20, 58F23

DOI:
https://doi.org/10.1090/S0002-9947-1993-1182980-3

MathSciNet review:
1182980

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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.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1993-1182980-3

Keywords:
Julia set,
exponential map,
model,
Cantor bouquet,
hairy arc

Article copyright:
© Copyright 1993
American Mathematical Society