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Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra II

Published online by Cambridge University Press:  22 April 2021

ALINE CERQUEIRA
Affiliation:
IMPA, Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina, Jardim Botânico, Rio de Janeiro, RJ, CEP 22460-320, Brazil (e-mail: alineagc@gmail.com)
CARLOS G. MOREIRA
Affiliation:
IMPA, Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina, Jardim Botânico, Rio de Janeiro, RJ, CEP 22460-320, Brazil
SERGIO ROMAÑA
Affiliation:
UFRJ, Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos 149, Centro de Tecnologia (Bloco C), Cidade Universitária, Ilha do Fundão, Rio de Janeiro, RJ, CEP 21941-909, Brazil (e-mail: sergiori@im.ufrj.br)
Corresponding

Abstract

Let $g_0$ be a smooth pinched negatively curved Riemannian metric on a complete surface N, and let $\Lambda _0$ be a basic hyperbolic set of the geodesic flow of $g_0$ with Hausdorff dimension strictly smaller than two. Given a small smooth perturbation g of $g_0$ and a smooth real-valued function f on the unit tangent bundle to N with respect to g, let $L_{g,\Lambda ,f}$ (respectively $M_{g,\Lambda ,f}$) be the Lagrange (respectively Markov) spectrum of asymptotic highest (respectively highest) values of f along the geodesics in the hyperbolic continuation $\Lambda $ of $\Lambda _0$. We prove that for generic choices of g and f, the Hausdorff dimensions of the sets $L_{g,\Lambda , f}\cap (-\infty , t)$ vary continuously with $t\in \mathbb {R}$ and, moreover, $M_{g,\Lambda , f}\cap (-\infty , t)$ has the same Hausdorff dimension as $L_{g,\Lambda , f}\cap (-\infty , t)$ for all $t\in \mathbb {R}$.

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Original Article
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Cerqueira, A., Matheus, C. and Moreira, C. G.. Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra. J. Mod. Dyn. 12 (2018), 151174.CrossRefGoogle Scholar
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Moreira, C. G.. Geometric properties of images of Cartesian products of regular Cantor sets by differentiable real maps. Preprint, 2016, arXiv:1611.00933.Google Scholar
Moreira, C. G. and Romaña, S.. On the Lagrange and Markov dynamical spectra for geodesic flows in surfaces with negative curvature. Preprint, 2015, arXiv:1505.05178.Google Scholar
Moreira, C. G. and Yoccoz, J.-C.. Tangences homoclines stables pour des ensembles hyperboliques de grande dimension fractale. Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), 168.CrossRefGoogle Scholar

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