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

Abstract

AbstractThe 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.