On the iterations of certain maps \(X \mapsto K \cdot(X + X^{- 1})\) over finite fields of odd characteristic
This work is concerned with the dynamics of rational maps of the form \(k\cdot(x+x^{-1})\) over finite fields of odd characteristic. The dynamics is studied by considering the graph whose vertices are the elements of the field, and where an edge joins two vertices if the dynamics carries one vertex into the other. The author identifies a set of values of \(k\) (depending on the characteristic) for which a detailed description of such graphs is possible, extending a previous investigation for the case \(k=1\) over fields of characteristic 2, 3 and 5 [the author, ibid. 133, No. 4, 1207--1228 (2013; Zbl 1306.37102); Contemp. Math. 579, 187--204 (2012; Zbl 1302.37074)].
scientific article; zbMATH DE number 6335922
scientific article