Hyperbolic components of rational maps: Quantitative equidistribution and counting

2017/05/15 by Thomas Gauthier, Gauthier, Thomas, Yûsuke Okuyama +3 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1705.05276

openalex publication_date 2017/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Λ be a quasi-projective variety and assume that, either Λ is a subvariety of the moduli space M_d of degree d rational maps, or Λ parametrizes an algebraic family (f_λ)_λ∈Λ of degree d rational maps on ℙ¹. We prove the equidistribution of parameters having p distinct neutral cycles towards the p-th bifurcation current letting the periods of the cycles go to ∞, with an exponential speed of convergence. We deduce several fundamental consequences of this result on equidistribution and counting of hyperbolic components. A key step of the proof is a locally uniform version of the quantitative approximation of the Lyapunov exponent of a rational map by the log⁺ of the modulus of the multipliers of periodic points.

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• Rigidity in Complex Dynamics: Multiplier Spectrum and Dynamical André-Oort Conjecture
• Analyticity of the Hausdorff dimension and metric structures on Misiurewicz families of polynomials
• Blow-up of multipliers in meromorphic families of rational maps

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