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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Infinite-dimensional Jacobi matrices associated with Julia sets


Authors: M. F. Barnsley, J. S. Geronimo and A. N. Harrington
Journal: Proc. Amer. Math. Soc. 88 (1983), 625-630
MSC: Primary 30D05; Secondary 33A65, 58F11
Erratum: Proc. Amer. Math. Soc. 92 (1984), 156.
MathSciNet review: 702288
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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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DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0702288-6
PII: S 0002-9939(1983)0702288-6
Article copyright: © Copyright 1983 American Mathematical Society