Infinite-dimensional Jacobi matrices associated with Julia sets

Barnsley, M. F.; Geronimo, J. S.; Harrington, A. N.

doi:10.1090/S0002-9939-1983-0702288-6

Infinite-dimensional Jacobi matrices associated with Julia sets
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by M. F. Barnsley, J. S. Geronimo and A. N. Harrington PDF

Proc. Amer. Math. Soc. 88 (1983), 625-630 Request permission

Erratum: Proc. Amer. Math. Soc. 92 (1984), 156.

Abstract:

Let $B$ be the Julia set associated with the polynomial $Tz = {z^N} + {k_1}{z^{N - 1}} + \cdots + {k_N}$, and let $\mu$ 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 $\mu$. The specific example $Tz = {z^3} - \lambda z$ is given, and some of the asymptotic properties of the entries in $J$ are presented.

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Mémoire sur l’iteration des fonctions rationelles

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