Infinite-Dimensional Jacobi Matrices Associated with Julia Sets

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