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

DOI:
https://doi.org/10.1090/S1088-4173-08-00183-5

Published electronically:
October 14, 2008

MathSciNet review:
2448263

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Abstract | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S1088-4173-08-00183-5

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.