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A Cr unimodal map with an arbitrary fast growth of the number of periodic points

Published online by Cambridge University Press:  19 April 2011

V. KALOSHIN  and
O. S. KOZLOVSKI
V. KALOSHIN
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA (email: kaloshin@math.psu.edu)
O. S. KOZLOVSKI
Affiliation:
University of Warwick, Coventry, CV4 7AL, England (email: oleg@maths.warwick.ac.uk)
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Abstract

In this paper we present a surprising example of a Cr unimodal map of an interval f:I→I whose number of periodic points Pn(f)=∣{x∈I:fnx=x}∣ grows faster than any ahead given sequence along a subsequence nk=3k. This example also shows that ‘non-flatness’ of critical points is necessary for the Martens–de Melo–van Strien theorem [M. Martens, W. de Melo and S. van Strien. Julia–Fatou–Sullivan theory for real one-dimensional dynamics. Acta Math.168(3–4) (1992), 273–318] to hold.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 1 , February 2012 , pp. 159 - 165
Copyright
Copyright © Cambridge University Press 2011

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References

[AM]Artin, M. and Mazur, B.. On periodic orbits. Ann. of Math. (2) 81 (1965), 82–99.CrossRefGoogle Scholar
[E]Epstein, H.. Fixed points of composition operators. Proceedings of NATO Advanced Study Institute on Nonlinear Evolution, Italy. Plenum, New York, 1988, pp. 71–100.Google Scholar
[GST]Gonchenko, S., Shilnikov, L. and Turaev, D.. On models with non-rough Poincaré homoclinic curves. Phys. D 62 (1993), 1–14.CrossRefGoogle Scholar
[K1]Kaloshin, V.. An extension of the Artin–Mazur theorem. Ann. of Math. (2) 150 (1999), 729–741.CrossRefGoogle Scholar
[K2]Kaloshin, V.. Generic diffeomorphisms with superexponential growth of the number of periodic orbits. Comm. Math. Phys. 211(1) (2000), 253–271.CrossRefGoogle Scholar
[K3]Kaloshin, V.. Growth of the number of periodic points. Normal Forms, Bifurcations and Finiteness Problems in Differential Equations. Eds. Ilyashenko, Y. and Rousseau, C.. Kluwer, Dordrecht, 2004, pp. 355–385.CrossRefGoogle Scholar
[KS]Kaloshin, V. and Saprykina, M.. Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits. Discrete Contin. Dyn. Syst. 15(2) (2006), 611–640.CrossRefGoogle Scholar
[K4]Kozlovski, O. S.. The dynamics of intersections of analytical manifolds. Dokl. Akad. Nauk 323(5) (1992), 823–825 (Engl. transl. Russian Acad. Sci. Dokl. Math. 45(2) (1992), 425–427).Google Scholar
[L]Lanford III, O. E.. A computer-assisted proof of the Feigenbaum conjectures. Bull. Amer. Math. Soc. (N.S.) 6(3) (1982), 427–434.CrossRefGoogle Scholar
[Ma]Martens, M.. Periodic points of renormalization. Ann. of Math. (2) 147(3) (1998), 435–484.CrossRefGoogle Scholar
[MMS]Martens, M., de Melo, W. and van Strien, S.. Julia–Fatou–Sullivan theory for real one-dimensional dynamics. Acta Math. 168(3–4) (1992), 273–318.CrossRefGoogle Scholar
[S]Sullivan, D.. Bounds, quadratic differentials, and renormalization conjectures. American Mathematical Society Centennial Publications, vol. II (Providence, RI, 1988). American Mathematical Society, Providence, RI, 1992, pp. 417–466.Google Scholar