Construction of mappings with attracting cycles

In [1], two methods to construct polynomial mappings with periodic points are given with Lagrange interpolation and Newton interpolation, and a conjecture that such polynomial mappings with chaotic behaviors should be a “generalized primitive polynomial” is raised. In this paper, we additionally consider stability of periodic points and give a new method to construct polynomial mappings with attracting cycles or superstable cycles. Based on this construction, we show how to further construct a mapping which is not in polynomial forms but possesses the same periodicity. We also discuss properties of such polynomials with integer cycles. Finally, we point out a falsity in [1] and give counterexamples against the conjecture in [1].