2019/04/03 by Ivan Chio, Chio, Ivan, Roland K. W. Roeder +1 · 1 citation
Mathematics · #05C31 #37F45 #82B20 #Advanced Topology and Set Theory #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Functional Equations Stability Results #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.1904.02195
openalex publication_date 2019/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Associated to any finite simple graph \Γ is the chromatic polynomial\nP_\Γ(q) whose complex zeroes are called the chromatic zeros of \Γ.\nA hierarchical lattice is a sequence of finite simple graphs\n \Γn n=0^\∞ built recursively using a substitution rule\nexpressed in terms of a generating graph. For each n, let \μn denote the\nprobability measure that assigns a Dirac measure to each chromatic zero of\n\Γn. Under a mild hypothesis on the generating graph, we prove that the\nsequence \μn converges to some measure \μ as n tends to infinity. We\ncall \μ the limiting measure of chromatic zeros associated to\n \Γn n=0^\∞. In the case of the Diamond Hierarchical Lattice we\nprove that the support of \μ has Hausdorff dimension two.\n The main techniques used come from holomorphic dynamics and more specifically\nthe theories of activity/bifurcation currents and arithmetic dynamics. We prove\na new equidistribution theorem that can be used to relate the chromatic zeros\nof a hierarchical lattice to the activity current of a particular marked point.\nWe expect that this equidistribution theorem will have several other\napplications.\n