Return times of polynomials as meta-Fibonacci numbers

Emerson, Nathaniel

doi:10.1090/S1088-4173-08-00183-5

Return times of polynomials as meta-Fibonacci numbers
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by Nathaniel D. Emerson

Conform. Geom. Dyn. 12 (2008), 153-173

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

Published electronically: October 14, 2008

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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.

References

Bodil Branner, Puzzles and para-puzzles of quadratic and cubic polynomials, Complex dynamical systems (Cincinnati, OH, 1994) Proc. Sympos. Appl. Math., vol. 49, Amer. Math. Soc., Providence, RI, 1994, pp. 31–69. MR 1315533, DOI 10.1090/psapm/049/1315533
Bodil Branner and John H. Hubbard, The iteration of cubic polynomials. II. Patterns and parapatterns, Acta Math. 169 (1992), no. 3-4, 229–325. MR 1194004, DOI 10.1007/BF02392761
Joseph Callaghan, John J. Chew III, and Stephen M. Tanny, On the behavior of a family of meta-Fibonacci sequences, SIAM J. Discrete Math. 18 (2005), no. 4, 794–824. MR 2157827, DOI 10.1137/S0895480103421397
Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383, DOI 10.1007/978-1-4612-4364-9
Laura G. DeMarco and Curtis T. McMullen, Trees and the dynamics of polynomials, to appear in Ann. Sci. École Norm. Sup.
Adrien Douady and John Hamal Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 287–343. MR 816367, DOI 10.24033/asens.1491
François Dubeau, On $r$-generalized Fibonacci numbers, Fibonacci Quart. 27 (1989), no. 3, 221–229. MR 1002065
Nathaniel D. Emerson, Dynamics of polynomials whose Julia set is an area zero Cantor set, Ph. D. thesis, 2001.
Nathaniel D. Emerson, Dynamics of polynomials with disconnected Julia sets, Discrete Contin. Dyn. Syst. 9 (2003), no. 4, 801–834. MR 1975358, DOI 10.3934/dcds.2003.9.801
Nathaniel D. Emerson, A family of meta-Fibonacci sequences defined by variable-order recursions, J. Integer Seq. 9 (2006), no. 1, Article 06.1.8, 21. MR 2211161
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. 467–511. MR 1215974
Jan Kiwi, Rational rays and critical portraits of complex polynomials, Ph. D. thesis, 1997. Stony Brook IMS pre-print 97-15.
Mikhail Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math. 178 (1997), no. 2, 185–247, 247–297. MR 1459261, DOI 10.1007/BF02392694
Mikhail Lyubich and John Milnor, The Fibonacci unimodal map, J. Amer. Math. Soc. 6 (1993), no. 2, 425–457. MR 1182670, DOI 10.1090/S0894-0347-1993-1182670-0
E. P. Miles Jr., Generalized Fibonacci numbers and associated matrices, Amer. Math. Monthly 67 (1960), 745–752. MR 123521, DOI 10.2307/2308649
Ricardo Pérez-Marco, Degenerate conformal structures, manuscript, 1999.
Dennis Sullivan, Bounds, quadratic differentials, and renormalization conjectures, American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988) Amer. Math. Soc., Providence, RI, 1992, pp. 417–466. MR 1184622