Abstract
For realλ a correspondence is made between the Julia setB λ forz→(z−λ)2, in the hyperbolic case, and the set ofλ-chainsλ±√(λ±√(λ±..., with the aid of Cremer's theorem. It is shown how a number of features ofBλ can be understood in terms ofλ-chains. The structure ofB λ is determined by certain equivalence classes ofλ-chains, fixed by orders of visitation of certain real cycles; and the bifurcation history of a given cycle can be conveniently computed via the combinatorics ofλ-chains. The functional equations obeyed by attractive cycles are investigated, and their relation toλ-chains is given. The first cascade of period-doubling bifurcations is described from the point of view of the associated Julia sets andλ-chains. Certain “Julia sets” associated with the Feigenbaum function and some theorems of Lanford are discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. A. Baker, D. Bessis, and P. Moussa, A Family of Almost Periodic Schrödinger Operators, Los Alamos preprint (1983).
M. F. Barnsley, J. S. Geronimo, and A. N. Harrington, Orthogonal Polynomials Associated with Invariant Measures on Julia Sets,Bull. A.M.S. 7:381–384 (1982).
M. F. Barnsley, J. S. Geronimo, and A. N. Harrington, Some Treelike Julia Sets and Padé Approximants,Lett. Math. Phys. 7:279–286 (1983).
M. F. Barnsley, J. S. Geronimo, and A. N. Harrington, On the Invariant Sets of a Family of Quadratic Maps,Commun. Math. Phys. 88:479–501 (1983).
M. F. Barnsley, J. S. Geronimo, and A. N. Harrington, Infinite Dimensional Jacobi Matrices Associated with Julia Sets,Proc. A.M.S. 88:625–630 (1983); also, Almost Periodic Operators Associated with Julia Sets (Georgia Institute of Technology Preprint), submitted toCommun. Math. Phys. (1983).
M. F. Barnsley, J. S. Geronimo, and A. N. Harrington, Geometry Electrostatic Measure and Orthogonal Polynomials on Julia Sets for Polynomials, to appear inJ. Ergodic Theory Dyn. Syst. (1983).
M. F. Barnsley, J. S. Geronimo, and A. N. Harrington, Condensed Julia Sets, with an Application to a Fractal Lattice Model Hamiltonian, (Georgia Institute of Technology Preprint, 1983), to appearTrans. Am. Math. Soc. (1984).
M. F. Barnsley and A. N. Harrington, Moments of Balanced Measures on Julia Sets,Trans. A.M.S. 284:271–280 (1984).
J. Bellissard, D. Bessis, and P. Moussa, Chaotic States for Almost Periodic Schrödinger Operators,Phys. Rev. Lett. 49:701–704 (1982).
D. Bessis, M. L. Mehta, and P. Moussa, Orthogonal Polynomials on a Family of Cantor Sets and the Problem of Iteration of Quadratic Maps,Lett. Math. Phys. 6:123–140 (1982).
D. Bessis and P. Moussa, Orthogonality Properties of Iterated Polynomial Mappings,Commun. Math. Phys. 88:503–529 (1983).
P. Billingsley,Ergodic Theory and Information (Wiley, New York, 1965).
H. Brolin, Invariant Sets under Iteration of Rational Functions,Ark. Mat. 6:103–144 (1965).
P. Collet and J. Eckmann,Iterated Maps on the Interval as Dynamical Systems (Birkhauser, Basel, Boston, 1980).
J. H. Curry, L. Garnett, and D. Sullivan, On the Iteration of a Rational Function: Computer Experiments with Newton's Method. To appearCommun. Math. Phys. (1983).
P. Collet and C. Tresser, Itérations d'Endomorphismes et Groupe de Renormalisation,C.R. Acad. Sc. Paris 287A:577–580 (1978).
B. Derrida, A. Gervois, and Y. Pomeau, Universal Metric Properties of Bifurcations of Endomorphisms,J. Phys. A12:269 (1979); also Iteration of Endomorphisms of the Real Axis and Representations of Numbers,Ann. Inst. Henri-Poincaré 29:305 (1978).
B. Derrida, L. DeSeze, and C. Itzykison, Fractal Structure of Zeros in Hierarchical Models, Preprint, CENS-SACLAY, Paris (1983).
E. Domany, S. Alexander, D. Bensimon and L. P. Kadanoff, Solutions to the Schrödinger Equation on Some Fractal Lattices,Phys. Rev. B 28:3110–3123 (1983).
A. Douady and J. H. Hubbard, Itération des Polynômes Quadratiques Complexes,C.R. Acad. Sci. Paris 294:123–126 (1982); see also A. Douady, Seminaire Bourbaki 599,Asterique, Vol.105–106:39–63 (1983).
M. P. Fatou, Sur les Equations Fonctionnelles,Bull. Soc. Math. France 47:161–271 (1919); 48:33–94; 48:208–314.
H. Epstein and J. Lascoux, Analyticity Properties of the Feigenbaum Function,Commun. Math. Phys. 81:437–453 (1981).
M. Feigenbaum, Quantitative Universality for a Class of Nonlinear Transformations,J. Stat. Phys. 19:25–52 (1978).
M. Feigenbaum, presentation at “Chaos in Dynamical Systems” conference, University of Maryland, March 1983.
J. Guckenheimer, On the Bifurcation of Maps of the Interval,Inventiones Math. 39:165–178 (1977).
J. Guckenheimer, Sensitive Dependence to Initial Conditions for One-Dimensional Maps,Commun. Math. Phys. 70:133–160 (1979).
G. Julia, Mémoire sur l'Itération des Fonctions Rationelles,J. Math. Pures Appl. 4:47–245 (1918).
R.-J. De Jonkere, Convergence de l'Itération des Fonctions Rationelles, (1963) thesis: Université Catholique de Louvain, Faculté des Sciences Appliquées.
T. Li and J. A. Yorke, Period Three Implies Chaos,Am. Math. Monthly 82:985–992 (1975).
O. E. Lanford III, A Computer Assisted Proof of the Feigenbaum Conjectures,Bull. A.M.S. 6:427–434 (1982).
B. B. Mandelbrot, Fractal Aspects of the Iterationz→λz(1−z) for Complexλ andz. Non-linear Dynamics, ed. R. H. G. Helleman,Ann. N. Y. Acad. Sci. 357, 249–259 (1980).
J. Mallet-Paret and J. A. Yorke, Snakes: Oriented Families of Periodic Orbits, Their Sources, Sinks, and Continuation,J. Diff. Equations 43:419–450 (1982).
B. Mendelson,Introduction to Topology (Blackie and Son, Ltd., London, 1983).
P. J. Myrberg, Sur l'Itération des Polynomes Réels Quadratiques,J. Math. Pures Appl. Sér. 41:339–351 (1962).
M. Metropolis, M. L. Stein, and P. R. Stein, On Finite Limit Sets for Transformations of the Unit Inverval,J. Combinat. Theory 15:25–44 (1973).
J. Milnor and P. Thurston, On Iterated Maps of the Interval, Preprint, Princeton (1977).
R. Rammel, On the Nature of Eigenstates on Fractal Structures,Phys. Rev. B 28:4871–4873 (1983); see alsoJ. Phys. (Paris) 45:191–206 (1984).
D. Ruelle, Repellers for Real Analytic Maps,Ergod. Theory Dyn. Syst. 2:99–108 (1982).
A. N. Sharkovskii, Coexistence of Cycles of a Continuous Map of a Line into Itself,Ukr. Mat. Z. 16:61–71 (1964).
D. Sullivan, private communication (1982).
Author information
Authors and Affiliations
Additional information
Supported by NSF grant No. MCS-8104862.
Supported by NSF grant No. MCS-8203325.
Rights and permissions
About this article
Cite this article
Barnsley, M.F., Geronimo, J.S. & Harrington, A.N. Geometry and combinatorics of Julia sets of real quadratic maps. J Stat Phys 37, 51–92 (1984). https://doi.org/10.1007/BF01012905
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01012905