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

Akbari, Monireh; Rabii, Maryam

doi:10.22061/jdma.2024.11165.1084

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

Document Type : Full Length Article

Authors

Monireh Akbari ¹
Maryam Rabii ²

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

10.22061/jdma.2024.11165.1084

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