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The geometry of Julia sets. (English) Zbl 0809.54034

Summary: 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.

MSC:

37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
37E05 Dynamical systems involving maps of the interval
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
54H20 Topological dynamics (MSC2010)
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