Summary
The Mahler measure formula expresses the height of an algebraic number as the integral of the log of the absolute value of its minimal polynomial on the unit circle. The height is in fact the canonical height associated to the monomial maps xn. We show in this work that for any rational map ϕ(x) the canonical height of an algebraic number with respect to ϕ can be expressed as the integral of the log of its equation against the invariant Brolin-Lyubich measure associated to ϕ, with additional adelic terms at finite places of bad reduction. We give a complete proof of this theorem using integral models for each iterate of ϕ. In the last chapter on equidistribution and Julia sets we give a survey of results obtained by P. Autissier, M. Baker, R. Rumely and ourselves. In particular our results, when combined with techniques of diophantine approximation, will allow us to compute the integrals in the generalized Mahler formula by averaging on periodic points.
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Pineiro, J., Szpiro, L., Tucker, T.J. (2005). Mahler measure for dynamical systems on ℙ1 and intersection theory on a singular arithmetic surface. In: Bogomolov, F., Tschinkel, Y. (eds) Geometric Methods in Algebra and Number Theory. Progress in Mathematics, vol 235. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4417-2_10
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DOI: https://doi.org/10.1007/0-8176-4417-2_10
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