×

Function weighted measures and orthogonal polynomials on Julia sets. (English) Zbl 0635.42021

Let T(z) be a monic polynomial of degree \(d\geq 2\) chosen so that its Julia set J is real. A class of invariant measures supported on J is constructed and discussed. We then construct the Jacobi matrices associated with these measures and show that they satisfy a renormalization group equation, a special case of which was discovered by Bellissard. Finally, we examine the asymptotic behavior of the orthogonal polynomials associated with these operators. We note that the operators have singular continuous spectra.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
30C10 Polynomials and rational functions of one complex variable
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. Avron, B. Simon (1982):Singular continuous spectrum for a class of almost periodic Jacobi matrices. Bull. Amer. Math. Soc.,6:81–85. · Zbl 0491.47014
[2] G. Baker, D. Bessis, P. Moussa (1984):A family of almost periodic Schrödinger operators. Physics,124A:61–78. · Zbl 0598.47054
[3] M. F. Barnsley, S. Demko (1985):Iterated function systems and the global construction of fractals. Proc. Roy. Soc. London Ser. A,399:243–275. · Zbl 0588.28002
[4] M. F.Barnsley, S. G.Demko, J. H.Elton, J. S.Geronimo (submitted):Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities. Ann. Inst. H. Poincaré. · Zbl 0653.60057
[5] M. F. Barnsley, J. S. Geronimo, A. N. Harrington (1984):Geometry, electrostatic measure and orthogonal polynomials on Julia sets for polynomials. Ergodic Theory Dynamical Systems,3:509–520. · Zbl 0566.41033
[6] M. F. Barnsley, J. S. Geronimo, A. N. Harrington (1985):Almost periodic Jacobi matrices associated with Julia sets. Comm. Math. Phys.,99:303–317. · Zbl 0574.58016
[7] M. F. Barnsley, J. S. Geronimo, A. N. Harrington (1985):Condensed Julia sets, with an application to a fractal lattice model Hamiltonian. Trans. Amer. Math. Soc.,288:537–561. · Zbl 0565.30018
[8] M. F. Barnsley, J. S. Geronimo, A. N. Harrington (1984):Geometrical and electrical properties of some Julia sets. In: Classical and Quantum Models and Arithmetic Problems (D. Chudnovsky, and G. Chudnovsky, eds.). Lecture Notes in Pure and Applied Mathematics, vol. 92.Marcel Decker, New York: pp. 1–68. · Zbl 0548.30021
[9] J.Bellissard (1985):Stability and instability in quantum mechanics. In: Trends in the Eighties (Ph. Blanchard, ed.). Singapore. · Zbl 0584.35024
[10] J. Bellissard, D. Bessis, P. Moussa (1982):Chaotic states of almost periodic Schrödinger operators. Phys. Rev. Lett.,49:701–704.
[11] D. Bessis, M. L. Mehta, P. Moussa (1982):Orthogonal polynomials on a family of Cantor sets and the problem of iterations of quadratic mappings. Lett. Math. Phys.,6:123–140. · Zbl 0483.33006
[12] D. Bessis, P. Moussa (1983):Orthogonality properties of iterated polynomial mappings. Comm. Math. Phys.,88:503–529. · Zbl 0523.30019
[13] P. Billingsley (1965): Ergodic Theory and Information. New York: Wiley. · Zbl 0141.16702
[14] H. Brolin (1965):Invariant sets under iteration of rational functions. Ark. Mat.,6:103–144. · Zbl 0127.03401
[15] A. Douady (1984–1985): Etude Dynamique des Polynômes Complexes. Orsay: Publications Mathematiques d’Orsay. · Zbl 0552.30018
[16] N. Dunford, J. Schwartz (1957): Linear Operators (Part I). New York: Wiley. · Zbl 0128.34803
[17] P. Fatou (1919, 1920):Sur les equations fonctionnelles. Bull. Soc. Math. France,47:161–271,48:33–94. · JFM 47.0921.02
[18] J. S.Geronimo (to appear):On the spectra of infinite-dimensional Jacobi matrices. J. Approx. Theory.
[19] J.Herndon (1985): Limit Periodicity of Sequences Defined by Certain Recurrence Relations and Julia Sets. Ph.D. Thesis, Georgia Institute of Technology.
[20] G. Julia (1918):Memoire sur l’iteration des fonctions rationnelles. J.Math. Pures Appl.,8:47–245. · JFM 46.0520.06
[21] A.Máté, P.Nevai, V.Totik (to appear):Strong and weak convergence of orthogonal polynomials. Amer. J. Math.
[22] P. Moussa (1986):Iteration des polynomes et proprietes d’orthogonalite. Ann. Inst. H. Poincaré,44:315–325. · Zbl 0607.30024
[23] P. Nevai (1979):Orthogonal polynomials. Mem. Amer. Math. Soc.,213:1–185. · Zbl 0405.33009
[24] G. Szegö (1975): Orthogonal Polynomials, 4th ed. American Mathematical Society Colloquium Publications, vol. 23. Providence, RI: AMS. · Zbl 0305.42011
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.