Return times of polynomials as metaFibonacci numbers
Author:
Nathaniel D. Emerson
Journal:
Conform. Geom. Dyn. 12 (2008), 153173
MSC (2000):
Primary 37F10, 37F50; Secondary 11B39
Published electronically:
October 14, 2008
MathSciNet review:
2448263
Fulltext PDF Free Access
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 metaFibonacci 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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 Bodil Branner and John H. Hubbard, Iteration of cubic polynomials, part II: patterns and parapatterns, Acta Math. 169 (1992), 229325. MR 1194004 (94d:30044)
 [CCT]
 Joseph Callaghan, John J. Chew, III, and Stephen M. Tanny, On the behavior of a family of metaFibonacci sequences, SIAM J. Discrete Math. 18 (2004), no. 4, 794824. MR 2157827 (2006c:11012)
 [CG]
 Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, SpringerVerlag, 1993. MR 1230383 (94h:30033)
 [DeMc]
 Laura G. DeMarco and Curtis T. McMullen, Trees and the dynamics of polynomials, to appear in Ann. Sci. École Norm. Sup.
 [DH]
 Adrien Douady and John Hamal Hubbard, On the dynamics of polynomiallike mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 287343. MR 816367 (87f:58083)
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 François Dubeau, On generalized Fibonacci numbers, Fibonacci Quart. 27 (1989), no. 3, 221229. MR 1002065 (90g:11022).
 [E1]
 Nathaniel D. Emerson, Dynamics of polynomials whose Julia set is an area zero Cantor set, Ph. D. thesis, 2001.
 [E2]
 , Dynamics of polynomials with disconnected Julia sets, Discrete Contin. Dyn. Syst. 9 (2003), no. 4, 801834. MR 1975358 (2004m:37083)
 [E3]
 , A family of metaFibonacci sequences defined by variableorder recursions, J. Integer Seq. 9 (2006), no. 1, Article 06.1.8, 21 pp. (electronic). MR 2211161
 [H]
 J. H. Hubbard, Local connectivity of Julia sets and bifurcation loci: three theorems of J.C. Yoccoz, Topological Methods in Modern Mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, pp. 467511. MR 1215974 (94c:58172)
 [K]
 Jan Kiwi, Rational rays and critical portraits of complex polynomials, Ph. D. thesis, 1997. Stony Brook IMS preprint 9715.
 [L]
 Mikhail Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math. 178 (1997), no. 2, 185247, 247297. MR 1459261 (98e:58145)
 [LM]
 Mikhail Lyubich and John Milnor, The Fibonacci unimodal map, J. Amer. Math. Soc. 6 (1993), no. 2, 425457. MR 1182670 (93h:58080)
 [Mi]
 E. P. Miles, Jr., Generalized Fibonacci numbers and associated matrices, Amer. Math. Monthly 67 (1960), no. 8, 745752. MR 0123521 (23 A846).
 [PM]
 Ricardo PérezMarco, Degenerate conformal structures, manuscript, 1999.
 [S]
 Dennis Sullivan, Bounds, quadratic differentials, and renormalization conjectures, American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988), AMS, Providence, RI, 1992, pp. 417466. MR 1184622 (93k:58194)
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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/S1088417308001835
PII:
S 10884173(08)001835
Keywords:
Julia set,
metaFibonacci,
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.
