Infinite-dimensional Jacobi matrices associated with Julia sets

Abstract

Let B B be the Julia set associated with the polynomial T z = z N + k 1 z N − 1 + ⋯ + k N Tz = {z^N} + {k_1}{z^{N - 1}} + \cdots + {k_N} , and let μ \mu be the balanced T T -invariant measure on B B . Assuming B B is totally real, we give relations among the entries in the infinite-dimensional Jacobi matrix J J whose spectral measure is μ \mu . The specific example T z = z 3 − λ z Tz = {z^3} - \lambda z is given, and some of the asymptotic properties of the entries in J J are presented.