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Mahler measure for dynamical systems on $$\mathbb P^1$$ and intersection theory on a singular arithmetic surface. (English) Zbl 1101.11020
Bogomolov, Fedor (ed.) et al., Geometric methods in algebra and number theory. Basel: Birkhäuser (ISBN 0-8176-4349-4/hbk). Progress in Mathematics 235, 219-250 (2005).
Let $$\phi\:\mathbb P^1\to\mathbb P^1$$ be a finite morphism, and let $$h_\phi$$ denote the corresponding canonical height. Let $$x$$ be an algebraic number and let $$F$$ be its minimal polynomial over $$\mathbb Z$$. This paper shows that the height $$h_\phi(x)$$ can be computed using the integral of the logarithm of the absolute value of $$F$$, using the invariant measure associated to $$\phi$$, as defined by H. Brolin [Ark. Mat. 6, 103–144 (1965; Zbl 0127.03401)]. Additional terms of a similar nature are also needed for finite places of bad reduction.
This generalizes the following well-known facts about the usual (Weil) logarithmic height $$h(x)$$: It satisfies the equation $$h(x^2)=2h(x)$$; the points of height zero are exactly the preperiodic points under the map $$x\mapsto x^2$$; these points are exactly the roots of unity, together with $$0$$ and $$\infty$$; the set of accumulation points of this set is the unit circle; the Mahler measure is an integral along the unit circle; and the Mahler measure of $$F$$ expresses the Weil height.
In the last section, the authors propose a conjecture that their integral at infinite places can be computed via equidistribution.
For the entire collection see [Zbl 1076.11001].

##### MSC:
 11G50 Heights 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets