On two affine-like dynamical systems in a local field (Q455800)

The paper consists of two parts. The first of them is devoted to the dynamical system on a local field corresponding to the mapping \(x\mapsto x^{p^n}+a\) where \(a\) is a fixed element of the field. This setting extends, on the one hand, the case of monomial dynamical systems [\textit{A. Yu. Khrennikov} and \textit{M. Nilsson}, \(p\)-adic deterministic and random dynamics. Dordrecht: Kluwer (2004; Zbl 1135.37003)] and, on the other hand, the case of affine dynamical systems [\textit{A.-H. Fan} and \textit{Y. Fares}, Arch. Math. 96, No. 5, 423--434 (2011; Zbl 1214.11134)]. The authors study minimal subsets with respect to this dynamical system, which happen to be cycles, and give their complete description.NEWLINENEWLINEIn the second part, the authors consider the case of a local field of a positive characteristic (more specifically, the case of a finite place of the corresponding global field) and define a dynamical system indexed not by natural numbers, as in all the existing papers on \(p\)-adic dynamics, but by elements of the ring of integers. The definition involves the Carlitz module and reflects the number-theoretic nature of the objects. For this case too, a description of orbits and minimal sets is given; properties of subsystems induced on minimal sets are studied.