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

Part of: Diophantine approximation, transcendental number theory Classical measure theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  22 April 2021

ALINE CERQUEIRA ,
CARLOS G. MOREIRA  and
SERGIO ROMAÑA Open the ORCID record for SERGIO ROMAÑA [Opens in a new window]
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: [email protected])
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: [email protected])
*
e-mail: [email protected]
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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 $t\in \mathbb {R}$ .

Keywords

flows on 3-manifoldsHausdorff dimensionhorseshoesLagrange spectrumMarkov spectrum

MSC classification

Primary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 11J06: Markov and Lagrange spectra and generalizations
Secondary: 28A78: Hausdorff and packing measures
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 6 , June 2022 , pp. 1898 - 1907
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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), 151–174.CrossRefGoogle Scholar
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