zbMATH — the first resource for mathematics

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].

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