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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.

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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