On the discontinuities of Hausdorff dimension in generic dynamical Lagrange spectrum

2024/03/27 by Christian Camilo Silva Villamil, Villamil, Christian Camilo Silva
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2403.18940

openalex publication_date 2024/03/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let φ₀ be a C²-conservative diffeomorphism of a compact surface S and let Λ₀ be a mixing horseshoe of φ₀. Given a smooth real function f defined in S and some diffeomorphism φ, close to φ₀, let L_{φ, f} be the Lagrange spectrum associated to the hyperbolic continuation Λ(φ) of the horseshoe Λ₀ and f. We show that, for generic choices of φ and f, if L_{φ, f} is the map that gives the Hausdorff dimension of the set L_{φ, f}∩ (-∞, t) for t∈ ℝ, then there are at most two points that can be limit of a infinite sequence of discontinuities of L_{φ, f}.

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