Function weighted measures and orthogonal polynomials on Julia sets. (English) Zbl 0635.42021

Let T(z) be a monic polynomial of degree \(d\geq 2\) chosen so that its Julia set J is real. A class of invariant measures supported on J is constructed and discussed. We then construct the Jacobi matrices associated with these measures and show that they satisfy a renormalization group equation, a special case of which was discovered by Bellissard. Finally, we examine the asymptotic behavior of the orthogonal polynomials associated with these operators. We note that the operators have singular continuous spectra.


42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
30C10 Polynomials and rational functions of one complex variable
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