Condensed Julia sets, with an application to a fractal lattice model Hamiltonian

Barnsley, M. F.; Geronimo, J. S.; Harrington, A. N.

doi:10.1090/S0002-9947-1985-0776392-7

Condensed Julia sets, with an application to a fractal lattice model Hamiltonian
HTML articles powered by AMS MathViewer

by M. F. Barnsley, J. S. Geronimo and A. N. Harrington PDF

Trans. Amer. Math. Soc. 288 (1985), 537-561 Request permission

Abstract:

The Julia set for the complex rational map $z \to {z^2} - \lambda + \varepsilon /z$, where $\lambda$ and $\varepsilon$ are complex parameters, is considered in the limit as $\varepsilon \to 0$. The result is called the condensed Julia set for $z \to ({z^3} - \lambda z)/z$. The limit of balanced measures, associated functional equations and orthogonal polynomials are considered; it is shown, for example, that for $\lambda \geqslant 2$ the moments, orthogonal polynomials, and associated Jacobi matrix $\mathcal {J}$ can be calculated explicitly and are not those belonging to ${z^2} - \lambda$. The spectrum of $\mathcal {J}$ consists of a point spectrum $P$ together with its derived set. The latter is the Julia set for ${z^2} - \lambda$, and carries none of the spectral mass when $\lambda > 2$. When $\lambda = 2$, $P$ is dense in $[-2,2]$. A similar condensation in the case $\lambda = 15/4$ leads to a system which corresponds precisely to the spectrum and density of states of a two-dimensional Sierpinski gasket model Schrödinger equation. The basic ideas about condensation of Julia sets in general are described. If $R(z)$ is a rational transformation of degree greater than one, then condensation can be attached to \[ z \to R(z) + \varepsilon \sum \limits _{i = 1}^k {{{(z - {a_i})}^{ - {\gamma _i}}},} \] where the ${\gamma _i}$’s and $k$ are finite positive integers and the ${a_i}$’s are complex numbers. If $\infty$ is an indifferent or attractive fixed point of $R(z)$, then all of the moments of the associated condensed balanced measure can be calculated explicitly, as can the orthogonal polynomials when the condensed Julia set is real. Sufficient conditions for the condensed measure $\sigma$ to be a weak limit of the balanced measures ${\mu _\varepsilon }$ are given. Functional equations connected to the condensed measure are derived, and it is noted that their form typifies those encountered in statistical physics, in connection with partition functions for Ising hierarchical models.

References

R. L. Adler and T. J. Rivlin, Ergodic and mixing properties of Chebyshev polynomials, Proc. Amer. Math. Soc. 15 (1964), 794–796. MR 202968, DOI 10.1090/S0002-9939-1964-0202968-3

The classical moment problem

Joseph Avron and Barry Simon, Almost periodic Schrödinger operators. I. Limit periodic potentials, Comm. Math. Phys. 82 (1981/82), no. 1, 101–120. MR 638515

A family of almost periodic Schródinger operators

M. F. Barnsley, J. S. Geronimo, and A. N. Harrington, Orthogonal polynomials associated with invariant measures on Julia sets, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 381–384. MR 663789, DOI 10.1090/S0273-0979-1982-15043-1
M. F. Barnsley, J. S. Geronimo, and A. N. Harrington, On the invariant sets of a family of quadratic maps, Comm. Math. Phys. 88 (1983), no. 4, 479–501. MR 702565
M. F. Barnsley, J. S. Geronimo, and A. N. Harrington, Infinite-dimensional Jacobi matrices associated with Julia sets, Proc. Amer. Math. Soc. 88 (1983), no. 4, 625–630. MR 702288, DOI 10.1090/S0002-9939-1983-0702288-6
M. F. Barnsley, J. S. Geronimo, and A. N. Harrington, Geometry, electrostatic measure and orthogonal polynomials on Julia sets for polynomials, Ergodic Theory Dynam. Systems 3 (1983), no. 4, 509–520. MR 753919, DOI 10.1017/S0143385700002108

Geometrical and electrical properties of some Julia sets

37

M. F. Barnsley, J. S. Geronimo, and A. N. Harrington, Some tree-like Julia sets and Padé approximants, Lett. Math. Phys. 7 (1983), no. 4, 279–286. MR 719458, DOI 10.1007/BF00420176

Almost periodic operators associated with Julia sets

M. F. Barnsley and A. N. Harrington, Moments of balanced measures on Julia sets, Trans. Amer. Math. Soc. 284 (1984), no. 1, 271–280. MR 742425, DOI 10.1090/S0002-9947-1984-0742425-6

Almost periodic Hamiltonians

an algebraic approach

J. Béllissard, D. Bessis, and P. Moussa, Chaotic states of almost periodic Schrödinger operators, Phys. Rev. Lett. 49 (1982), no. 10, 701–704. MR 669364, DOI 10.1103/PhysRevLett.49.701

Renormalization group calculations of finite systems

order parameter and specific heat for epitaxial ordering

12

D. Bessis, J. S. Geronimo, and P. Moussa, Mellin transforms associated with Julia sets and physical applications, J. Statist. Phys. 34 (1984), no. 1-2, 75–110. MR 739123, DOI 10.1007/BF01770350
Daniel Bessis, Madan Lal Mehta, and Pierre Moussa, Polynômes orthogonaux sur des ensembles de Cantor et itérations des transformations quadratiques, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), no. 15, 705–708 (French, with English summary). MR 650541
D. Bessis, M. L. Mehta, and P. Moussa, Orthogonal polynomials on a family of Cantor sets and the problem of iterations of quadratic mappings, Lett. Math. Phys. 6 (1982), no. 2, 123–140. MR 651127, DOI 10.1007/BF00401737
D. Bessis and P. Moussa, Orthogonality properties of iterated polynomial mappings, Comm. Math. Phys. 88 (1983), no. 4, 503–529. MR 702566
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
Hans Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103–144 (1965). MR 194595, DOI 10.1007/BF02591353
T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 0481884

Dynamics of

On balanced measures

B. Derrida, L. de Seze, and C. Itzykson, Fractal structure of zeros in hierarchical models, J. Statist. Phys. 33 (1983), no. 3, 559–569. MR 732376, DOI 10.1007/BF01018834
B. Derrida, J.-P. Eckmann, and A. Erzan, Renormalisation groups with periodic and aperiodic orbits, J. Phys. A 16 (1983), no. 5, 893–906. MR 712597
B. Derrida and H. J. Hilhorst, Singular behaviour of certain infinite products of random $2\times 2$ matrices, J. Phys. A 16 (1983), no. 12, 2641–2654. MR 715727

Lattices of effectively non-integral dimensionality

18

Eytan Domany, Shlomo Alexander, David Bensimon, and Leo P. Kadanoff, Solutions to the Schrödinger equation on some fractal lattices, Phys. Rev. B (3) 28 (1983), no. 6, 3110–3123. MR 717348, DOI 10.1103/physrevb.28.3110
Adrien Douady, Systèmes dynamiques holomorphes, Bourbaki seminar, Vol. 1982/83, Astérisque, vol. 105, Soc. Math. France, Paris, 1983, pp. 39–63 (French). MR 728980
Adrien Douady and John Hamal Hubbard, Itération des polynômes quadratiques complexes, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 3, 123–126 (French, with English summary). MR 651802
P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France 47 (1919), 161–271 (French). MR 1504787
Ricardo Mañé, On the uniqueness of the maximizing measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), no. 1, 27–43. MR 736567, DOI 10.1007/BF02584743
Lisl Gaal, Classical Galois theory with examples, Markham Publishing Co., Chicago, Ill., 1971. MR 0280465
John Guckenheimer, Endomorphisms of the Riemann sphere, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 95–123. MR 0274740

Exemples de fractions rationelles ayant une orbite dense sur la sphere de Riemann

Memoire sur l’iteration des fonctions rationelles

4

Exactly soluble Ising models on hierarchical lattices

24

Benoit B. Mandelbrot, The fractal geometry of nature, Schriftenreihe für den Referenten. [Series for the Referee], W. H. Freeman and Co., San Francisco, Calif., 1982. MR 665254

Fractal aspects of the iteration of

357

Ricardo Mañé, On the uniqueness of the maximizing measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), no. 1, 27–43. MR 736567, DOI 10.1007/BF02584743

On the dynamics of rational maps

David R. Nelson and Michael E. Fisher, Soluble renormalization groups and scaling fields for low-dimensional Ising systems, Ann. Physics 91 (1975), 226–274. MR 391850, DOI 10.1016/0003-4916(75)90284-5
Tom S. Pitcher and John R. Kinney, Some connections between ergodic theory and the iteration of polynomials, Ark. Mat. 8 (1969), 25–32. MR 263125, DOI 10.1007/BF02589532
R. Rammal, Nature of eigenstates on fractal structures, Phys. Rev. B (3) 28 (1983), no. 8, 4871–4874. MR 720978, DOI 10.1103/physrevb.28.4871
R. Rammal, Spectrum of harmonic excitations on fractals, J. Physique 45 (1984), no. 2, 191–206. MR 737523, DOI 10.1051/jphys:01984004502019100

Ergodic rational maps with dense critical point-forward orbit

David Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems 2 (1982), no. 1, 99–107. MR 684247, DOI 10.1017/s0143385700009603
Dennis Sullivan, Itération des fonctions analytiques complexes, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 9, 301–303 (French, with English summary). MR 658395

Solution of the Fatou-Julia problem on wandering domains

Structural stability implies hyperbolicity for Kleinian groups

Topological conjugacy classes of analytic endomorphisms

Orthogonal polynomials