## 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

### Keywords:

heights; entropy; morphic height

### Citations:

Zbl 0826.14015; Zbl 0895.14006
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