Keller, Karsten Symbolic dynamics for angle-doubling on the circle. III: Sturmian sequences and the quadratic map. (English) Zbl 0830.58011 Ergodic Theory Dyn. Syst. 14, No. 4, 787-805 (1994). It is known that the structure of Julia sets of quadratic maps is tightly connected with the angle-doubling map \(h\) on the circle \(T\). A connected and locally connected Julia set can be considered as a topological factor \(T/_\approx\) of \(T\) with respect to a special \(h\)-invariant equivalence relation \(\approx\) on \(T\), called by the author Julia equivalence. It was proved before by the author that a map \(\alpha\) from \(T\) onto the set of all Julia equivalences gives a natural abstract description of the Mandelbrot set.A topological classification of the abstract Julia sets \(T/_{\approx_\alpha}\) is given too. It is proved that \(T/_{\approx_\alpha}\) contains simple closed curves if the point \(\alpha\) has a periodic kneading sequence. The set of points having a periodic kneading sequence is characterized and the relation of this set to Julia sets and to Mandelbrot set is discussed.[For parts I and II 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)]. Reviewer: G.V.Khmelevskaja-Plotnikova (Namur) Cited in 1 ReviewCited in 2 Documents MSC: 37E99 Low-dimensional dynamical systems Keywords:symbolic dynamics; angle-doubling set; circle; Julia sets; Mandelbrot set; quadratic map; Sturmian sequences; Julia equivalence; kneading sequence Citations:Zbl 0768.58013; Zbl 0785.58021; Zbl 0830.58012 PDFBibTeX XMLCite \textit{K. Keller}, Ergodic Theory Dyn. Syst. 14, No. 4, 787--805 (1994; Zbl 0830.58011) Full Text: DOI References: [1] DOI: 10.1088/0951-7715/6/3/003 · Zbl 0785.58021 [2] Branner, Proc. Symp. Appl. Math. 39 pp 75– (1989) [3] DOI: 10.1002/mana.19911540104 · Zbl 0824.28007 [4] Atela, Ergod. Th. & Dynam. Sys. 12 pp 401– (1991) [5] Schröder, Fractals, Chaos, Power Laws (1991) [6] Lyubich, Usp. Mat. Nauk. 41 pp 35– (1986) [7] DOI: 10.2307/2371431 · Zbl 0022.34003 [8] Lavaurs, C. R. Acad. Sci. 303 pp 143– (1986) [9] Keller, Topology, Measures, and Fractals pp 76– (1992) [10] DOI: 10.1512/iumj.1981.30.30055 · Zbl 0598.28011 [11] Falconer, Fractal Geometry (1990) [12] Douady, Topological Methods in Modern Mathematics (1993) [13] Douady, Chaotic Dynamics and Fractals pp 155– (1986) [14] DOI: 10.1007/BF01762232 · Zbl 0256.54028 [15] Blanchard, Fractal Geometry and Analysis 346 (1989) [16] Beardon, Iteration of Rational Functions (1992) [17] Bandt, Springer Lecture Notes in Mathematics 1514 pp 1– (1992) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.