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Symbolic dynamics for angle-doubling on the circle. IV: Equivalence of abstract Julia sets. (English) Zbl 0830.58012

For each locally connected Julia set \(J_c\) corresponding to a quadratic map \(p_c\) on the circle \(T\), one finds a Julia equivalence \(\approx\) such that \(p_c\) on the concrete Julia set \(J_c\) and the factor map \(h/_\approx\) on the “abstract” Julia set \(T/_\approx\) are topologically conjugate \((h\) is the angle-doubling set on \(T)\).
The aim of the work is to find by a simple property on the points \(\alpha, \gamma \in T\) when \((T/_{\approx_\alpha},h/_{\approx_\alpha})\) and \((T/_{\approx_\gamma},h_{\approx_\gamma})\) are topologically conjugate.
[For parts I and II of this paper see C. Bandt and the author, Lect. Notes Math. 1514, 1-23 (1992; Zbl 0768.58013) and Nonlinearity 6, No. 3, 377-392 (1993; Zbl 0785.58021), for part III see the review above].

MSC:

37E99 Low-dimensional dynamical systems
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