Keller, Karsten Symbolic dynamics for angle-doubling on the circle. IV: Equivalence of abstract Julia sets. (English) Zbl 0830.58012 Atti Semin. Mat. Fis. Univ. Modena 42, No. 2, 547-567 (1994). 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]. Reviewer: G.V.Khmelevskaja-Plotnikova (Namur) Cited in 1 Review MSC: 37E99 Low-dimensional dynamical systems Keywords:symbolic dynamics; angle-doubling set; circle; Julia equivalence; Julia set; topologically conjugate Citations:Zbl 0830.58011; Zbl 0768.58013; Zbl 0785.58021 PDFBibTeX XMLCite \textit{K. Keller}, Atti Semin. Mat. Fis. Univ. Modena 42, No. 2, 547--567 (1994; Zbl 0830.58012)