An algorithm for counting the number of periodic points of a family of polynomials

Document Type : Full Length Article

Authors

1 Department of Mathematics, Faculty of Basic Sciences, Shahid Rajaee Teacher Training University, Tehran, Iran

2 Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University,Tehran, Iran

Abstract

In this paper we consider the family fa(x) = axd(x − 1) + x when
a < 0 is a real number and d ≥ 2 is an even integer. The function fa has a
unique positive critical point. By decreasing the parameter a, the behavior of
the orbit of this critical point changes. In this paper we consider two cases. In
the first case the orbit of the positive critical point converges to 0 and in the
second case the positive critical point is mapped to a repelling periodic point
of period 2. In each case we give a recursive formula to determine the number
of the periodic points of fa. Also, in each case we introduce an invariant set
on which fa is chaotic. We employ conjugacy map and symbolic dynamics in
our investigations.

Graphical Abstract

An algorithm for counting the number of periodic points of a family of polynomials

Keywords

Main Subjects

References

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Journal of Discrete Mathematics and Its Applications
Volume 9, Issue 4
December 2024
Pages 249-267

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History

  • Receive Date: 24 October 2024
  • Revise Date: 05 November 2024
  • Accept Date: 19 November 2024
  • Publish Date: 01 December 2024

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