Concentration of dimension in the Lagrange spectrum

2024/11/25 by Christian Camilo Silva Villamil, Villamil, Christian Camilo Silva
Engineering · #Dynamical Systems (math.DS) #FOS: Mathematics #Field-Flow Fractionation Techniques #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2411.16939

openalex publication_date 2024/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let φ be a smooth conservative diffeomorphism of a compact surface S and let Λ be a mixing horseshoe of φ. Given a smooth real function f defined on S, we define for points η in the unstable Cantor set of the pair (φ,Λ), a generalization, k_φ,Λ,f(η), of the best constant of Diophantine approximation for irrational numbers. We study the set of points η for which the sets k_φ,Λ,f^-1((-∞,η]) and k_φ,Λ,f^-1(η) have the same Hausdorff dimension and when the Hausdorff dimension of Λ is less than one, we describe generically the local Hausdorff dimension of the dynamical Lagrange spectrum, L_φ,Λ,f, restricted to this set of points. Finally, we recover the same results for the classical Lagrange spectra.

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