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Conformal Geometry and Dynamics
Conformal Geometry and Dynamics
ISSN 1088-4173

 

Return times of polynomials as meta-Fibonacci numbers


Author: Nathaniel D. Emerson
Journal: Conform. Geom. Dyn. 12 (2008), 153-173
MSC (2000): Primary 37F10, 37F50; Secondary 11B39
Published electronically: October 14, 2008
MathSciNet review: 2448263
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Abstract: We consider generalized closest return times of a complex polynomial of degree at least two. Most previous studies on this subject have focused on the properties of polynomials with particular return times, especially the Fibonacci numbers. We study the general form of these closest return times. The main result of this paper is that these closest return times are meta-Fibonacci numbers. In particular, this result applies to the return times of a principal nest of a polynomial. Furthermore, we show that an analogous result holds in a tree with dynamics that is associated with a polynomial.


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Additional Information

Nathaniel D. Emerson
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089
Email: nemerson@usc.edu

DOI: http://dx.doi.org/10.1090/S1088-4173-08-00183-5
PII: S 1088-4173(08)00183-5
Keywords: Julia set, meta-Fibonacci, polynomial, principal nest, puzzle, return time, tree with dynamics.
Received by editor(s): December 10, 2007
Published electronically: October 14, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.