Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Authors: M. F. Barnsley, J. S. Geronimo and A. N. Harrington
Journal: Trans. Amer. Math. Soc. 288 (1985), 537-561
MSC: Primary 58F11; Secondary 30D05, 82A68
DOI: https://doi.org/10.1090/S0002-9947-1985-0776392-7
MathSciNet review: 776392
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

$\displaystyle 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 [Enhancements On Off] (What's this?)

  • [AR] R. L. Adler and T. J. Rivlin, Ergodic and mixing properties of Chebychev polynomials, Proc. Amer. Math. Soc. 15 (1964), 794-796. MR 0202968 (34:2827)
  • [Ak] N. I. Akhiezer, The classical moment problem, Hafner, New York, 1965.
  • [AS] J. Avron and B. Simon, Almost periodic Schródinger operators, Comm. Math. Phys. 82 (1981), 101-120. MR 638515 (84i:34023)
  • [BaBM] G. A. Baker, D. Bessis and P. Moussa, A family of almost periodic Schródinger operators, Los Alamos Preprint, 1981.
  • [BGH1] 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), 381-384. MR 663789 (84a:42031)
  • [BGH2] -, On the invariant sets of a family of quadratic maps, Comm. Math. Phys. 88 (1983), 479-501. MR 702565 (85g:58055)
  • [BGH3] -, Infinite dimensional Jacobi matrices associated with Julia sets, Proc. Amer. Math. Soc. 88 (1983), 625-630. MR 702288 (85a:30040)
  • [BGH4] -, Geometry electrostatis measure and orthogonal polynomials on Julia sets for polynomials, J. Ergodic Theory Dynamical Systems 3 (1983), 509-520. MR 753919 (86a:58052)
  • [BGH5] -, Geometrical and electrical properties of some Julia sets, J. Statist. Phys. 37 (1984), 51-92.
  • [BGH6] -, Some treelike Julia sets and Padé approximants, Lett. Math. Phys. 7 (1983), 279-286. MR 719458 (84m:41026)
  • [BGH7] -, Almost periodic operators associated with Julia sets, Preprint, Georgia Institute of Technology, 1983; Comm. Math. Phys. (to appear).
  • [BH] M. F. Barnsley and A. N. Harrington, Moments of balanced measures on Julia sets, Trans. Amer. Math. Soc. 284 (1984), 271-280. MR 742425 (85j:30053)
  • [BT] J. Bellissard and D. Testard, Almost periodic Hamiltonians: an algebraic approach, Preprint, C.N.R.S., Marseille, France, July 1981.
  • [BBM] J. Bellissard, D. Bessis and P. Moussa, Chaotic states for almost periodic Schródinger operators, Phys. Rev. Lett. 49 (1982), 701-704. MR 669364 (83k:58052)
  • [Bo] A. N. Berker and S. Ostlund, Renormalization group calculations of finite systems: order parameter and specific heat for epitaxial ordering, J. Phys. C: Solid State Phys. 12 (1979), 4961-4975.
  • [BGM] D. Bessis, J. S. Gerónimo and P. Moussa, Mellin transforms associated with Julia sets and physical applications, J. Statist. Phys. 34 (1984), 15-110. MR 739123 (85i:58067)
  • [BMM1] D. Bessis, M. L. Mehta and P. Moussa, Polynômes orthogonaux sur des ensembles de Cantor et iterations des transformations quadratiques, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), 705-708. MR 650541 (84e:58036)
  • [BMM2] -, Orthogonal polynomials on a family of Cantor sets and the problem of iteration of quadratic maps, Lett. Math. Phys. 6 (1982), 123-140. MR 651127 (83g:58038)
  • [BM] D. Bessis and P. Moussa, Orthogonality properties of iterated polynomial mappings, Comm. Math. Phys. 88 (1983), 503-529. MR 702566 (85a:58053)
  • [B1] P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. (N.S.) 11 (1984), 85-141. MR 741725 (85h:58001)
  • [Br] H. Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103-144. MR 0194595 (33:2805)
  • [C] T. S. Chihara, An introduction to orthogonal polynomials, Gordon & Breach, New York, 1978. MR 0481884 (58:1979)
  • [DK] R. L. Devaney and M. Krych, Dynamics of $ \exp (z)$, Preprint, Boston University, 1983.
  • [D] S. Demko, On balanced measures (unpublished notes).
  • [DDI] B. Derrida, L. De Seze and C. Itzykison, Fractal structure of zeros in hierarchical models, J. Statist. Phys. 33 (1983), 559-569. MR 732376 (84m:82112)
  • [DEE] B. Derrida, J. P. Eckmann and A. Erzan, Renormalization groups with periodic and aperiodic orbits, J. Phys. A: Math. Gen. 16 (1983), 893-906. MR 712597 (85d:82042)
  • [DH] B. Derrida and H. Hihlhorst, Singular behaviour of certain infinite products of random matrices, J. Phys. A: Math. Gen. 16 (1983), 2641-2654. MR 715727 (85h:82005)
  • [Dh] D. Dhar, Lattices of effectively non-integral dimensionality, J. Math. Phys. 18 (1977), 577-585.
  • [DABK] E. Domany, S. Alexander, D. Bensimon and L. P. Kadanoff, Solutions to the Schródinger equation on some fractal lattices, Phys. Rev. B 28 (1983), 3110-3123. MR 717348 (85h:82033)
  • [Do] A. Douady, Systèmes dynamiques holomorphes, Séminaire Bourbaki, 35e annéé, 1982/83, #599, 1982. MR 728980 (85h:58090)
  • [DoH] A. Douady and J. Hubbard, Itération des polynômes quadratiques complexés, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), 123-216. MR 651802 (83m:58046)
  • [F] M. P. Fatou, Sur les equations fonctionelles, Bull. Soc. Math. France 47 (1919), 161-271; ibid. 48, 33-94; ibid. 48, 208-314. MR 1504787
  • [FLM] A. Freire, A. Lopes and R. Mañè, An invariant measure for rational maps, Preprint, 1982. MR 736568 (85m:58110b)
  • [G] L. Gaal, Classical Galois theory with examples, Markham, Chicago, Ill., 1971. MR 0280465 (43:6185)
  • [Gu] J. Guckenheimer, Endomorphisms of the Riemann sphere, Proc. Sympos. Pure Math. vol. 14, Amer. Math. Soc., Providence, R. I., 1970, pp. 95-123. MR 0274740 (43:500)
  • [H] M. Herman, Exemples de fractions rationelles ayant une orbite dense sur la sphere de Riemann, Preprint, Ecole Polytechnique, 1983.
  • [J] G. Julia, Memoire sur l'iteration des fonctions rationelles, J. Math. Pures Appl. 4 (1918), 47-245.
  • [KG] M. Kaufman and R. B. Griffiths, Exactly soluble Ising models on hierarchical lattices, Phys. Rev. B 24 (1981), 496-498.
  • [M1] B. Mandlebrot, The fractal geometry of nature, Freeman, San Francisco, Calif., 1982. MR 665254 (84h:00021)
  • [M2] -, Fractal aspects of the iteration of $ z \to \lambda z(1 - z)$, Ann. N.Y. Acad. Sci. 357 (1980), 249-259.
  • [Ma] R. Mañè, On the uniqueness of the maximizing measure for rational maps, Preprint, Rio de Janeiro, 1982. MR 736567 (85m:58110a)
  • [MSS] R. Manñé, P. Sad and D. Sullivan, On the dynamics of rational maps, Preprint.
  • [NF] D. R. Nelson and M. E. Fisher, Soluble renormalization group and scaling fields for low dimensional Ising systems, Ann. Physics 91 (1975), 226-274. MR 0391850 (52:12669)
  • [PK] T. S. Pitcher and J. R. Kinney, Some connections between ergodic theory and iteration of polynomials, Ark. Mat. 8 (1968), 25-32. MR 0263125 (41:7730)
  • [R1] R. Rammal, On the nature of eigenstates on fractal structures, Phys. Rev. B 28 (1983), 4871-4874. MR 720978 (84k:82025)
  • [R2] -, Spectrum of harmonic excitations on fractals, J. Physique (Paris) 45 (1984), 191-206. MR 737523 (85d:82101)
  • [Re] M. Rees, Ergodic rational maps with dense critical point-forward orbit, University of Minnesota Mathematics Report, 82-140.
  • [Ru] D. Ruelle, Repellers for real analytic maps, Ergodic Theory Dynamical Systems 2 (1982), 99-108. MR 684247 (84f:58095)
  • [S1] D. Sullivan, Iteration des fonctions analytiques complexes, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), 301. MR 658395 (83d:58061)
  • [S2] -, Solution of the Fatou-Julia problem on wandering domains, Preprint, 1982.
  • [S3] -, Structural stability implies hyperbolicity for Kleinian groups, Preprint, 1982.
  • [S4] -, Topological conjugacy classes of analytic endomorphisms, Preprint, 1982.
  • [Sz] G. Szego, Orthogonal polynomials, Amer. Math. Soc., Providence, R.I., 1939.
  • [T] W. Thurston, Lecture Notes, CBMS Conf., Univ. of Minnesota, 1983.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F11, 30D05, 82A68

Retrieve articles in all journals with MSC: 58F11, 30D05, 82A68


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0776392-7
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society