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Morphic heights and periodic points. (English) Zbl 1048.11091

Chudnovsky, David (ed.) et al., Number theory: New York seminar 2003. New York, NY: Springer (ISBN 0-387-40655-7/hbk), 167-177 (2004).
Let \(\phi :{\mathbb P}^1(\overline{{\mathbb Q}})\to {\mathbb P}^1(\overline{{\mathbb Q}})\) be a morphism of degree \(d\) defined over \({\mathbb Q}\). G. S. Call and J. H. Silverman resp. G. S. Call and S. W. Goldstine [Compos. Math. 89, 163–205 (1993; Zbl 0826.14015), J. Number Theory 63, 211–243 (1997; Zbl 0895.14006)] proved that there exists a so-called canonical global morphic height \(\hat{\lambda}_{\phi}\) on \(\overline{{\mathbb Q}}\) having the following properties: (i) \(\hat{\lambda}_{\phi}(q)= d\hat{\lambda}_{\phi}(q)\) for \(q\in{\mathbb P}^1(\overline{{\mathbb Q}})\); (ii) \(\hat{\lambda}_{\phi}(q)=0\) if and only if \(q\) is pre-periodic, i.e., if the orbit \(\{ \phi^{(n)}(q):\, n=1,2,\ldots\}\) is finite, where \(\phi^{(n)}\) is the \(n\)-th iterate of \(\phi\). This global height can be decomposed into a sum of local heights \(\hat{\lambda}_{\phi}(q)=\sum_v n_v\lambda_{\phi ,v}(q)\) where the sum is over all infinite and finite places of the number field \(K:={\mathbb Q}(q)\) and \(n_v=[K_v:{\mathbb Q}_v]/[K:{\mathbb Q}]\).
In this paper, the authors restrict themselves to the case that \(\phi [x,y] =[y^df(x/y) ,y^d]\) where \(f\) is a polynomial in \({\mathbb Q}[X]\) of degree \(d\). In this case, they derive for each \(v\) an expression for \(\lambda_{\phi ,v}(q)\) comparable to Jensen’s formula for the Mahler measure. Roughly speaking the expression gives the average of the logarithmic distance to the topological closure of the set of periodic points of \(\phi\). At the end of the paper the authors consider in more detail the case that \(f\) is a Chebyshev polynomial.
For the entire collection see [Zbl 1030.00027].

MSC:

37P30 Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems
11G50 Heights
11S05 Polynomials
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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