Periodic points, multiplicities, and dynamical units.

*(English)*Zbl 0813.11059Let \(K\) 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.

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.

Reviewer: J.H.Silverman (Providence)

##### MSC:

11R09 | Polynomials (irreducibility, etc.) |

11R27 | Units and factorization |

37B99 | Topological dynamics |

26A18 | Iteration of real functions in one variable |

12E99 | General field theory |