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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
MathSciNet review: 776392
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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.

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