2015/02/16 by Niki Myrto Mavraki, Mavraki, Niki Myrto, Hexi Ye +1
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Number Theory (math.NT) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1502.04660
openalex publication_date 2015/02/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Baker-Rumely and Favre-Rivera-Letelier independently proved an important arithmetic equidistribution theorem for points of small height on the Berkovich compactification of the projective line with respect to an adelic measure on ℙ1. Around the same time, Chambert-Loir proved a more general version of this arithmetic equidistribution theorem in the setting of curves from a different approach. We generalize the notion of an adelic measure to that of a quasi-adelic measure on ℙ1, and show that arithmetic equidistribution of points with small height holds for quasi-adelic measures as well. Moreover, we show that the canonical measure associated with a dynamical pair (f,c) on ℙ1 is rarely adelic. We prove that for certain examples of families of rational functions parameterized by ℙ1, corresponding to the curve Per1(λ) introduced by Milnor for a root of unity λ, the measure corresponding to a general starting point is quasi-adelic. Finally, we place our results in context by establishing their connection with two problems in arithmetic dynamics.