On two affine-like dynamical systems in a local field. (English) Zbl 1273.37046

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 [A. Yu. Khrennikov and 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 [A.-H. Fan and 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.
In 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.


37P20 Dynamical systems over non-Archimedean local ground fields
11S82 Non-Archimedean dynamical systems
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[1] Briend, J.-Y.; Benedetto, R.; Perdry, H., Dynamique des polynômes quadratiques sur LES corps locaux, J. théor. nr. bordx., 19.1, 325-336, (2005) · Zbl 1197.37144
[2] Chabert, J.-L.; Fan, A.-H.; Fares, Y., Minimal dynamical systems on a discrete valuation domain, Discrete contin. dyn. syst., 25, 777-795, (2009) · Zbl 1185.37014
[3] Fan, A.-H.; Fares, Y., Minimal subsystems of affine dynamics on local fields, Arch. math., 96, 423-434, (2011) · Zbl 1214.11134
[4] Fares, Y., Factorial preservation, Arch. math., 83, 497-506, (2004) · Zbl 1073.13011
[5] Goss, D., Basic structures of function field arithmetic, Ergeb. math. grenzgeb. (3), vol. 35, (1996), Springer-Verlag Berlin · Zbl 0874.11004
[6] Khrennikov, A., Non-Archimedean analysis, quantum paradoxes, dynamical system and biological models, (1997), Kluwer · Zbl 0920.11087
[7] Khrennikov, A.; Nilsson, M., P-adic deterministic and random dynamics, (2004), Kluwer Dordrecht
[8] Kochubei, A., Analysis in positive characteristic, Cambridge tracts in math., vol. 178, (2009), Cambridge University Press · Zbl 1171.12005
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