Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra

2016/02/15 by Cerqueira, Aline, Matheus, Carlos, Moreira, Carlos Gustavo
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1602.04649

Abstract

Let φ₀ be a smooth area-preserving diffeomorphism of a compact surface M and let Λ₀ be a horseshoe of φ₀ with Hausdorff dimension strictly smaller than one. Given a smooth function f:M→ ℝ and a small smooth area-preserving perturtabion φ of φ₀, let L_{φ, f}, resp. M_{φ, f} be the Lagrange, resp. Markov spectrum of asymptotic highest, resp. highest values of f along the φ-orbits of points in the horseshoe Λ obtained by hyperbolic continuation of Λ₀. We show that, for generic choices of φ and f, the Hausdorff dimension of the sets L_{φ, f}∩ (-∞, t) vary continuously with t∈ℝ and, moreover, M_{φ, f}∩ (-∞, t) has the same Hausdorff dimension of L_{φ, f}∩ (-∞, t) for all t∈ℝ.

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