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Symbolic dynamics for angle-doubling on the circle. I: The topology of locally connected Julia sets. (English) Zbl 0768.58013

Ergodic theory and related topics III, Proc. 3rd Int. Conf., Güstrow/Ger. 1990, Lect. Notes Math. 1514, 1-23 (1992).
[For the entire collection see Zbl 0744.00035.]
The authors study the dynamics of quadratic polynomials viewed as complex dynamical systems in the complex plane.
Using ideas of Douady, Hubbard and Thurston the topology of the Julia set \(J_ c\) of a polynomial \(p_ c(z)=z^ 2+c\) is studied in the case that \(J_ c\) is locally connected and \(c\) has an external ray. The most important tools used in the present paper are: itineraries, kneading sequences, shift spaces and laminations of the unit disc.
The Julia set turns out to be the quotient space of the unit circle by points with equal itineraries. Special cases like Siegel disc case, periodic case and tree-like Julia sets are discussed in detail.
Reviewer: H.Kriete (Aachen)

MSC:

37E99 Low-dimensional dynamical systems
37B99 Topological dynamics
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable

Citations:

Zbl 0744.00035
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