On cycles and orbits of polynomial mappings \(\mathbb{Z}^2\mapsto\mathbb{Z}^2\). (English) Zbl 1057.11016

Pezda, T.

Acta Math. Inform. Univ. Ostrav. 10, No. 1, 95-102 (2002).

Let \({\mathbb Z}_2\) be the ring of \(2\)-adic integers. The author proves that the orders of periodic points of mappings \({\mathbb Z}_2^2\rightarrow {\mathbb Z_2^2}\) defined by polynomials with coefficients in \({\mathbb Z}_2\) lie in the set \(\Omega=\{1,2,3,4,6,8,9,12, 16,18,24\}\), and for every \(n\in\Omega\) there is such a mapping with a periodic point of order \(n\). Then he applies the Hasse principle for periodic points, proved by him on other place [Acta Arith. 108, 127–146 (2003; Zbl 1020.11066)], to deduce that the set of possible orders of periodic points of \(2\)-dimensional polynomial mappings \({\mathbb Z^2}\rightarrow {\mathbb Z}^2\) also equals \(\Omega\).

Reviewer: Władysław Narkiewicz (Wrocław)

MSC:

11C08	Polynomials in number theory
11S05	Polynomials
14E05	Rational and birational maps
37C25	Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics

Keywords:

polynomial cycles; finite polynomial orbits; polynomial maps

Citations:

Zbl 1020.11066

References:

[1]	Narkiewicz W., Pezda T.: Finite polynomial orbits in finitely generated domains. Mh. Math. 124, 309-316 (1997). · Zbl 0897.13024
[2]	Pezda T.: Polynomial cycles in certain local domains. Acta Arith., LXVI. 1, 1994, 11-22 · Zbl 0803.11063
[3]	Pezda T.: Cycles of polynomial mappings in several variables. Manuscr. Math., 83, 1994, 279-289. · Zbl 0804.11059
[4]	Pezda T.: Cycles of polynomial mappings in several variables over rings of integers in finite extensions of rationals. Acta Arith., to appear. · Zbl 1197.37143