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

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 \(\Omega\).


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


Zbl 1020.11066
Full Text: EuDML


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.