proceedings of the american mathematical society Volume 88. Number 4. August 1983 INFINITE-DIMENSIONAL JACOBI MATRICES ASSOCIATEDWITH JULIA SETS M. F. BARNSLEY1. J. S. GERÓNIMO2 AND A. N. HARRINGTON Abstract. Let B be the Julia set associated with the polynomial Tz = z N + k¡z s ~ ' + •■ ■ +A v. and let ¡i be the balanced T-invariant measure on B. Assuming B is totally real, we give relations among the entries in the infinite-dimensional Jacobi matrix J whose spectral measure is ¡i. The specific example Tz = r' — Xz is given, and some of the asymptotic properties of the entries in J are presented. 1. Introduction. Let C he the complex plane and T: C -» C a polynomial, T(z) = zN + kxzN~x + ■■■ +kN where N > 2 and each k, E C. Define T°(z) = z and T"(z) = T ° T"~x(z) for « G {1,2,3_}. A fundamental role in the study of the sequence of iterates {T"(z)) is played by the Julia set B. B is the set of points z E C where {T"(z)} is not normal in the sense of Montel, and a general exposition can be found in Julia [8], Fatou [6,7] and Brolin [5], It has positive logarithmic capacity, and on it can be placed an equilibrium charge distribution p. This provides a measure on B which is invariant under T: B -» B and is such that the system (B,p, T) is strongly mixing. In an earlier paper [1] we investigated general properties of ju and its associated orthogonal monic polynomials. Here we restrict attention to the case where B is a compact subset of the real line, and the orthogonal polynomials satisfy a three-term recurrence formula. In [2] we proved, for N = 2, relationships connecting the coefficients, which permit all the polynomials to be calculated in a recursive fashion. Here we generalized the relationships so that the orthogonal polynomials of all degrees can be obtained for any T for which B is a compact subset of the real line (Theorem 1). The results are illustrated for T(z) — z3 — Xz with X 3= 3. When X = 3 the polynomials are those of Chebychev, shifted to the interval [-2,2], and when X > 3 they become a generalization whose support is a Cantor set. In this case we establish that both the coefficients (Theorem 2) and the associated Jacobi matrix / (Theorem 3) display an asymptotic self-reproducing property. 2. Preliminaries. Definition 1. p is a balanced T-invariant Borel measure on B if p is a probability measure supported on B, such that for any complete assignment of branches of T~l, namely Tfx forj E (1,2,3,.. .,JV}, p(Tfx(S)) = p(S)/N for each Borel set S. Received by the editors March 23, 1982. 1980 Mathematics Subject Classification.Primary 30C10; Secondary 47B25. 'Supported by NSF grant MCS-8104862. Supported by NSF grant MCS-8002731. il 983 American Mathematical Society 0002-9947/82/0000-1166/S02.O0 625 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use