On two affine-like dynamical systems in a local field
type: Item
instance of: scholarly article
title: On two affine-like dynamical systems in a local field
zbMATH Open document ID: 1273.37046
author: David Adam
author: Youssef Fares
published in: Journal of Number Theory
publication date: 2012-10-22T00:00:00Z
reviewed by: Anatoly N. Kochubei
Mathematics Subject Classification ID: 37P20
Mathematics Subject Classification ID: 11S82
zbMATH DE Number: 6097269
zbMATH Keywords: dynamical system
zbMATH Keywords: local field
zbMATH Keywords: minimal subset
zbMATH Keywords: Carlitz module
MaRDI profile type: Publication
full work available at URL: j.jnt.2012.05.011
OpenAlex ID: W2084724161
cites work: Dynamics of quadratic polynomials over local fields
cites work: Minimal dynamical systems on a discrete valuation domain
cites work: Minimal subsystems of affine dynamics on local fields
cites work: Factorial preservation
cites work: Q4714153
cites work: Q4211216
cites work: Q4658084
cites work: Q3615516
DOI: 10.1016/J.JNT.2012.05.011
review text: 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.
description: scientific article; zbMATH DE number 6097269