Continuity of Hausdorff Dimension Across Generic Dynamical Lagrange and Markov Spectra II

2017/11/10 by A. Cerqueira, Cerqueira, A., C. G. Moreira +3
Mathematics · #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1711.03851

openalex publication_date 2017/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let g₀ be a smooth pinched negatively curved Riemannian metric on a complete surface N, and let Λ₀ be a basic hyperbolic set of the geodesic flow of g₀ with Hausdorff dimension strictly smaller than two. Given a small smooth perturbation g of g₀ and a smooth real-valued function f on the unit tangent bundle to N with respect to g, let L_g,Λ,f, resp. M_g,Λ,f be the Lagrange, resp. Markov spectrum of asymptotic highest, resp. highest values of f along the geodesics in the hyperbolic continuation Λ of Λ₀. We prove that, for generic choices of g and f, the Hausdorff dimension of the sets L_{g,Λ, f}∩ (-∞, t) vary continuously with t∈ℝ and, moreover, M_{g,Λ, f}∩ (-∞, t) has the same Hausdorff dimension of L_{g,Λ, f}∩ (-∞, t) for all t∈ℝ.

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