2004/07/25 by Matthew Baker, Baker, Matthew, Robert Rumely +1 · 3 citations
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Number Theory (math.NT) #advanced mathematical theories #math.DS #math.NT
paper · pdf · doi:10.48550/arxiv.math/0407426
50 pages; v2 contains additional references, exposition has been modified, and Sections 7 and 8 from v1 have been removed to shorten the paper's length
openalex publication_date 2004/07/25 · arxiv created 2005/07/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If phi(z) is a rational function on P1 of degree at least 2 with coefficients in a number field k, we compute the homogeneous transfinite diameter of the v-adic filled Julia sets of phi for all places v of k by introducing a new quantity called the homogeneous sectional capacity. In particular, we show that the product over all places of these homogeneous transfinite diameters is 1. We apply this product formula and some new potential-theoretic results concerning Green's functions on Riemann surfaces and Berkovich spaces to prove an adelic equidistribution theorem for dynamical systems on the projective line. This theorem, which generalizes the results of Baker-Hsia, says that for each place v of k, there is a canonical probability measure on the Berkovich space P1Berk,v over Cv such that if zn is a sequence of algebraic points in P1 whose canonical heights with respect to phi tend to zero, then the zn's and their Galois conjugates are equidistributed with respect to muphi,v for all places v of k. For archimedean v, P1Berk,v is just the Riemann sphere, muphi,v is Lyubich's invariant measure, and our result is closely related to a theorem of Lyubich and Freire-Lopes-Mane.