## 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 $$\Omega$$.

### 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

Zbl 1020.11066
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### References:

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