Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-12T10:07:15.857Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometry, electrostatic measure and orthogonal polynomials on Julia sets for polynomials

Published online by Cambridge University Press:  19 September 2008

M. F. Barnsley,
J. S. Geronimo and
A. N. Harrington
M. F. Barnsley
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
J. S. Geronimo
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
A. N. Harrington
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

The Julia set B for an N'th degree polynomial T and its equilibrium electrostatic measure μ are considered. The unique balanced measure on B is shown to be μ. Integral properties of μ and of the monic polynomials orthogonal with respect to μ, Pn, n = 0, 1, 2, …, are derived. Formulae relating orthogonal polynomials of the second kind of different degrees are displayed. The measure μ is recovered both in the limit from the zeros and from the poles of the [Nn − 1/Nn] Padé approximant to the moment generating function to μ. For infinitely many polynomials of each degree N the zeros and poles all lie on an increasing sequence of trees of analytic arcs contained in B. The properties of these Padé approximant sequences support conjectures of George Baker which have not previously been tested on measures supported on sets nearly as complicated as Julia sets spread out in the complex plane.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 3 , Issue 4 , December 1983, pp. 509 - 520
Copyright
Copyright © Cambridge University Press 1983

References

REFERENCES

[1]Baker, G. A. & Graves-Morris, P. R.. Padé Approximants, (Two volumes). Encyclopedia of Mathematics and Its Applications. Addison-Wesley: 1981.Google Scholar
[2]Barnsley, M. F. & Harrington, A. N.. Moments of balanced measures on Julia sets. To appear in Trans. Amer. Math. Soc. (1983).Google Scholar
[3]Barnsley, M. F., Bessis, D. & Moussa, P.. The Diophantine moment problem and the analytic structure in the activity of the ferromagnetic Ising model. J. Math. Phys. 20 (1979), 535–546.CrossRefGoogle Scholar
[4]Barnsley, M. F., Geronimo, J. S. & Harrington, A. N.. On the invariant sets of a family of quadratic maps. (Original Aug. 1981.) Shortened version: Comm. Math. Phys. 88 (1983), 479–501.CrossRefGoogle Scholar
[5]Barnsley, M. F., Geronimo, J. S. & Harrington, A. N.. Orthogonal polynomials associated with invariant measures on Julia sets. Bull. Amer. Math. Soc. 7 (1982), 381–384.CrossRefGoogle Scholar
[6]Barnsley, M. F., Geronimo, J. S. & Harrington, A. N.. Some treelike Julia sets and Padé approximants. Letters in Math. Phys. 7 (1983), 279–286.CrossRefGoogle Scholar
[7]Barnsley, M. F., Geronimo, J. S. & Harrington, A. N.. Infinite dimensional Jacobi matrices associated with Julia sets. Proc. Amer. Math. Soc. 88 (1983), 625–630.CrossRefGoogle Scholar
[8]Barnsley, M. F., Geronimo, J. S. & Harrington, A. N.. Geometrical and electrical properties of some Julia sets. To appear in an anthology edited by G. V. and D. Chudnovsky.Google Scholar
[9]Bellissard, J., Bessis, D. & Moussa, P.. Chaotic states for almost periodic Shrödinger operators. Phys. Rev. Letters 49 (1982), 701–704.CrossRefGoogle Scholar
[10]Bessis, D. & Moussa, P.. Orthogonality properties of iterated polynomial mappings. Comm. Math. Phys. 88 (1983), 503–529.CrossRefGoogle Scholar
[11]Bessis, D., Mehta, M. L. & Moussa, P.. Orthogonal polynomials on a family of Cantor sets and the problem of iterations of quadratic mappings. Letters in Math. Phys. 6 (1982), 123–140.CrossRefGoogle Scholar
[12]Brolin, H.. Invariant sets under iteration of rational functions. Arkiv for Matematik 6, (1965), 103–144.CrossRefGoogle Scholar
[13]Collet, P. & Eckmann, J., Iterated Maps on the Interval as Dynamical Systems. Birkhauser: Basel-Boston, 1980.Google Scholar
[14]Demko, S.. In preparation.Google Scholar
[15]Fatou, M. P.. Sur les equations fonctionnelles. Bull. Soc. Math. de France 47 (1919), 161–271;Google Scholar
Fatou, M. P.. Sur les equations fonctionnelles. Bull. Soc. Math. 48, 33–94;Google Scholar
Fatou, M. P.. Sur les equations fonctionnelles. Bull. Soc. Math. 48, 208–314.Google Scholar
[16]Grebogi, C., Ott, E. & Yorke, J. A.. Chaotic attractors in crisis. Phys. Rev. Letters 48 (1982), 1507–1510.CrossRefGoogle Scholar
[17]Jakobson, M. V.. Absolutely continuous invariant measures for one parameter families of onedimensional maps. Comm. Math. Phys. 81 (1981), 39–88.CrossRefGoogle Scholar
[18]de Jonkere, R.-J.. Convergence de l'iteration des fonctions rationelles. Thesis, (1963), Université Catholique de Louvain, Faculté des Sciences Appliqueés.Google Scholar
[19]Julia, G.. Mémoire sur l'iteration des fonctions rationelles. Journal de Math. Pures et Appliqueés 4 (1918), 47–245.Google Scholar
[20]Rudin, W.. Real and Complex Analysis. McGraw-Hill: New York, 1966.Google Scholar
[21]Sullivan, D.. Iteration des fonctions analytiques complexes. C.R. Acad. Sci. Paris 294 (1982), 301–303.Google Scholar