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

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Kaloshin, V. and Kozlovski, O.. (2012) A Cr unimodal map with an arbitrary fast growth of the number of periodic points. Ergodic Theory and Dynamical Systems, Vol.32 (No.1). pp. 159-165. ISSN 0143-3857

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Official URL: http://dx.doi.org/10.1017/S0143385710000817

Abstract

In this paper we present a surprising example of a C(r) unimodal map of an interval f : I -> I whose number of periodic points P(n)(f) = vertical bar{x is an element of I : f(n) x = x}vertical bar grows faster than any ahead given sequence along a subsequence (n)k = 3(k). 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.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Mappings (Mathematics)
Journal or Publication Title:	Ergodic Theory and Dynamical Systems
Publisher:	Cambridge University Press
ISSN:	0143-3857
Date:	February 2012
Volume:	Vol.32
Number:	No.1
Page Range:	pp. 159-165
Identification Number:	10.1017/S0143385710000817
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	American Institute of Mathematics (AIM)
References:	[AM] M. Artin and B. Mazur. On periodic orbits. Ann. of Math. (2) 81 (1965), 82–99. [E] H. Epstein. Fixed points of composition operators. Proceedings of NATO Advanced Study Institute on Nonlinear Evolution, Italy. Plenum, New York, 1988, pp. 71–100. [GST] S. Gonchenko, L. Shilnikov and D. Turaev. On models with non-rough Poincaré homoclinic curves. Phys. D 62 (1993), 1–14. [K1] V. Kaloshin. An extension of the Artin–Mazur theorem. Ann. of Math. (2) 150 (1999), 729–741. [K2] V. Kaloshin. Generic diffeomorphisms with superexponential growth of the number of periodic orbits. Comm. Math. Phys. 211(1) (2000), 253–271. [K3] V. Kaloshin. Growth of the number of periodic points. Normal Forms, Bifurcations and Finiteness Problems in Differential Equations. Eds. Y. Ilyashenko and C. Rousseau. Kluwer, Dordrecht, 2004, pp. 355–385. [KS] V. Kaloshin and M. Saprykina. Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits. Discrete Contin. Dyn. Syst. 15(2) (2006), 611–640. [K4] O. S. Kozlovski. 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). [L] O. E. Lanford III. A computer-assisted proof of the Feigenbaum conjectures. Bull. Amer. Math. Soc. (N.S.) 6(3) (1982), 427–434. [Ma] M. Martens. Periodic points of renormalization. Ann. of Math. (2) 147(3) (1998), 435–484. [MMS] 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. [S] D. Sullivan. 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.
URI:	http://wrap.warwick.ac.uk/id/eprint/41916