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Periodic points, multiplicities, and dynamical units. (English) Zbl 0813.11059
Let $$K$$ be a number field with ring of integers $$R$$, and let $$\varphi (z) \in K(z)$$ be a rational function of degree $$d \geq 2$$. The authors use periodic points of $$\varphi$$ to construct algebraic units in number fields, analogous to the classical construction of cyclotomic and elliptic units. For example, suppose that $$\varphi (z) \in R[z]$$ is a monic polynomial. W. Narkiewicz [Colloq. Math. 58, 151-155 (1989; Zbl 0703.12002)] proved that if $$\alpha \in \overline K$$ is a point of exact period $$n$$ for $$\varphi$$ and if $$i,j$$ are integers with $$\text{gcd} (i-j,n) = 1$$, then $$(\varphi^ i (\alpha) - \varphi^ j (\alpha))/(\varphi (\alpha) - \alpha)$$ is an algebraic unit. In this setting the authors show that if $$\alpha_ 1$$ and $$\alpha_ 2$$ have exact periods $$n_ 1$$ and $$n_ 2$$ with $$n_ 1 \nmid n_ 2$$ and $$n_ 2 \nmid n_ 1$$, then $$\alpha_ 1 - \alpha_ 2$$ is an algebraic unit. They also generalize both of these results to arbitrary rational functions $$\varphi (z) \in K(z)$$. The tools used to prove these results are varied. In order to produce units from the orbit of a single periodic point (as Narkiewicz does), they define a $${\mathfrak p}$$-adic distance function and prove that if $$\varphi$$ has good reduction at $${\mathfrak p}$$, then it is $${\mathfrak p}$$-adically nonexpansive at every point. This result is also used to prove that if $$\varphi$$ has good separable reduction at $${\mathfrak p}$$, then it has only finitely many attracting periodic points.
The construction of units using points with different periods is done by looking at a cycle $$Z^*_ n(\varphi)$$ of points having essential (or formal) period $$n$$, where now $$\varphi \in k(z)$$ is an arbitrary algebraically closed field. The points appearing in $$Z^*_ n (\varphi)$$ have actual period dividing $$n$$, and the authors give a good description of which periodic points appear in terms of the multiplier of the periodic point and the characteristic of $$k$$. This information is combined with a reduction theorem which says that if $$\varphi$$ has good reduction at $${\mathfrak p}$$, then the reduction of $$Z^*_ n (\varphi)$$ modulo $${\mathfrak p}$$ is equal to $$Z^*_ n (\widetilde \varphi \text{mod} {\mathfrak p})$$.
In a final section the authors present several numerical examples illustrating their results and describe some multiplicative relations among the units that they produce.

##### MSC:
 11R09 Polynomials (irreducibility, etc.) 11R27 Units and factorization 37B99 Topological dynamics 26A18 Iteration of real functions in one variable 12E99 General field theory
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