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

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