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Symbolic dynamics for angle-doubling on the circle. II: Symbolic description of the abstract Mandelbrot set. (English) Zbl 0785.58021

Summary: Under the assumption that the Mandelbrot set is locally connected, its boundary can be considered as a topological factor \(T/\sim\) of the circle \(T\), which is called the abstract Mandelbrot set. The structure of \(T/\sim\) is tightly connected with the angle-doubling map on \(T\). We give an abstract description of the equivalence relation \(\sim\), discuss results by Thurston and Lavaurs from the viewpoint of symbolic dynamics and study renormalization in the abstract Mandelbrot set.
[For part I see the authors, Lect. Notes Math. 1514, 1-23 (1992; Zbl 0768.58013)].

MSC:

37E99 Low-dimensional dynamical systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37B99 Topological dynamics

Citations:

Zbl 0768.58013
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