The Mandelbrot set and $\sigma$-automorphisms of quotients of the shift
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Abstract:
In this paper we study how certain loops in the parameter space of quadratic complex polynomials give rise to shift-automorphisms of quotients of the set ${\Sigma _2}$ of sequences on two symbols. The Mandelbrot set ${\mathbf {M}}$ is the set of parameter values for which the Julia set of the corresponding polynomial is connected. Blanchard, Devaney, and Keen have shown that closed loops in the complement of the Mandelbrot set give rise to shift-automorphisms of ${\Sigma _2}$ , i.e., homeomorphisms of ${\Sigma _2}$ that commute with the shift map. We study what happens when the loops are not entirely in the complement of the Mandelbrot set. We consider closed loops that cross the Mandelbrot set at a single main bifurcation point, surrounding a component of ${\mathbf {M}}$ attached to the main cardioid. If $n$ is the period of this component, we identify a period- $n$ orbit of ${\Sigma _2}$ to a single point. The loop determines a shift-automorphism of this quotient space of ${\Sigma _2}$ . We give these maps explicitly.References
- Paul Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 1, 85–141. MR 741725, DOI 10.1090/S0273-0979-1984-15240-6
- Paul Blanchard, Robert L. Devaney, and Linda Keen, The dynamics of complex polynomials and automorphisms of the shift, Invent. Math. 104 (1991), no. 3, 545–580. MR 1106749, DOI 10.1007/BF01245090
- Mike Boyle, John Franks, and Bruce Kitchens, Automorphisms of one-sided subshifts of finite type, Ergodic Theory Dynam. Systems 10 (1990), no. 3, 421–449. MR 1074312, DOI 10.1017/S0143385700005678 Böttcher, Bull. Kasan Math. Soc. 14 (1905), 176.
- Bodil Branner, The Mandelbrot set, Chaos and fractals (Providence, RI, 1988) Proc. Sympos. Appl. Math., vol. 39, Amer. Math. Soc., Providence, RI, 1989, pp. 75–105. MR 1010237, DOI 10.1090/psapm/039/1010237
- Robert L. Devaney, An introduction to chaotic dynamical systems, 2nd ed., Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. MR 1046376
- Adrien Douady and John Hamal Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 287–343. MR 816367 —, Itération des polynômes quadratiques complexes, C. R. Acad. Sci. Paris Sér. I 294. —, Etude dynamique des polynômes complexes, Publ. Math. Orsay, Part I, No. 84-02, 1984, and Part II, No. 85-04, 1985.
- A. Douady, Algorithms for computing angles in the Mandelbrot set, Chaotic dynamics and fractals (Atlanta, Ga., 1985) Notes Rep. Math. Sci. Engrg., vol. 2, Academic Press, Orlando, FL, 1986, pp. 155–168. MR 858013 —, Julia sets and the Mandelbrot set, The Beauty of Fractals (H.-O. Peitgen and P. Richter, eds.), Springer-Verlag, 1986, pp. 161-173.
- P. Fatou, Sur l’itération des fonctions transcendantes Entières, Acta Math. 47 (1926), no. 4, 337–370 (French). MR 1555220, DOI 10.1007/BF02559517
- G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory 3 (1969), 320–375. MR 259881, DOI 10.1007/BF01691062 G. Julia, Mémoires sur l’iteration des fonctions rationelles, J. Math. Pures Appl. 8 (1915), 47-245.
- Robert L. Devaney and Linda Keen (eds.), Chaos and fractals, Proceedings of Symposia in Applied Mathematics, vol. 39, American Mathematical Society, Providence, RI, 1989. The mathematics behind the computer graphics; Lecture notes prepared for the American Mathematical Society Short Course held in Providence, Rhode Island, August 6–7, 1988; AMS Short Course Lecture Notes. MR 1010232, DOI 10.1090/psapm/039 P. Lavaurs, Une propriété de continuité, Expose No. 17, in [DH2].
- H.-O. Peitgen and P. H. Richter, The beauty of fractals, Springer-Verlag, Berlin, 1986. Images of complex dynamical systems. MR 852695, DOI 10.1007/978-3-642-61717-1
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 335 (1993), 683-703
- MSC: Primary 58F23; Secondary 58F03, 58F13
- DOI: https://doi.org/10.1090/S0002-9947-1993-1075379-1
- MathSciNet review: 1075379