A. N. Harrington

The calculus of fractal interpolation functions

Abstract The calculus of deterministic fractal functions is introduced. Fractal interpolation functions can be explicitly indefinitely integrated any number of times, yielding a hierarchy of successively smoother interpolation functions which generalize splines and which, just as in the case for the original fractal functions, are attractors for iterated function systems. The fractal dimension for a class of fractal interpolation functions is explicitly computed.

On the Invariant Sets of a Family of Quadratic Maps

The Julia setBλ for the mappingz → (z−λ)2 is considered, where λ is a complex parameter. For λ≧2 a new upper bound for the Hausdorff dimension is given, and the monic polynomials orthogonal with respect to the equilibrium measure onBλ are introduced. A method for calculating all of the polynomials is provided, and certain identities which obtain among coefficients of the three-term recurrence relations are given. A unifying theme is the relationship betweenBλ and λ-chains λ±√ (λ±√ (λ± ...), which is explored for −1/4≦λ≦2 and for λ∈ℂ with |λ|≦1/4, with the aid of the Böttcher equation. ThenBλ is shown to be a Hölder continuous curve for |λ|<1/4.

A Mandelbrot set for pairs of linear maps

The set of points A ( s )= {± 1 ± s ± s 2 ± s 3 ± … for all sequences of + and −}, generically a fractal. For example A (1/3) is the classical Cantor set and A (1/2 + i/2) is a dragon curve. This set of fractals can be classified in terms of an associated Mandelbrot set D = s ∈ : ¦s¦ A(s) is disconnected. The structure of D and its boundary are investigated.

Almost periodic Jacobi matrices associated with Julia sets for polynomials

AbstractLetℐ be the Jacobi matrix associated with polynomialT(z) of degreeN≧2. The spectrum ofℐ is the Julia set associated withT(z) which in many cases is a Cantor set. Letℐ(1) denote the result of omitting the first row and column ofJ. Then it is shown that the spectrum ofℐ(1) may be purely discrete.It is also shown that forT(z)=αNCN(z/α) for α>

Some tree-like Julia sets and Pad approximants

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Approximation Theory on a Snowflake

, whereCN is a Chebychev polynomial the coefficients ofℐ andℐ(1) are limit periodic extending the work of Bellissard, Bessis, and Moussa (Phys. Rev. Lett.49, 701–704 (1982)).

Julia Sets and Autonomous Differential Equations

For realλ a correspondence is made between the Julia setBλ forz→(z−λ)2, in the hyperbolic case, and the set ofλ-chainsλ±√(λ±√(λ±..., with the aid of Cremers theorem. It is shown how a number of features ofBλ can be understood in terms ofλ-chains. The structure ofBλ is determined by certain equivalence classes ofλ-chains, fixed by orders of visitation of certain real cycles; and the bifurcation history of a given cycle can be conveniently computed via the combinatorics ofλ-chains. The functional equations obeyed by attractive cycles are investigated, and their relation toλ-chains is given. The first cascade of period-doubling bifurcations is described from the point of view of the associated Julia sets andλ-chains. Certain “Julia sets” associated with the Feigenbaum function and some theorems of Lanford are discussed.

Orthogonal polynomials associated with invariant measures on Julia sets

AbstractThe Julia set B for T(z)=(z−λ)2, the equilibrium electrostatic measure μ on B, the associated orthogonal polynomials, Pn, and the Padé approximants to the moment-generating function for μ are considered. When 0≤λ≤2, the locations of the zeros and poles for the Padé approximant sequence