On the Lagrange and Markov dynamical spectra for geodesic flows on surfaces with negative curvature

Carlos Gustavo T. de A. Moreira; Sergio Augusto Romaña Ibarra

doi:10.3934/jmd.2023005

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2023, Volume 19: 187-236. Doi: 10.3934/jmd.2023005

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On the Lagrange and Markov dynamical spectra for geodesic flows on surfaces with negative curvature

1.
SUSTech International Center for Mathematics, Shenzhen, Guangdong, People's Republic of China
2.
Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, cep 22460-320, Rio de Janeiro, Brasil
3.
Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos 149, Instituto de Matématica, Centro de Tecnologia-Bloco C, Cidade Universitária Ilha do Fundão, cep 21941-909, Rio de Janeiro, Brasil

Received: June 17, 2021

Revised: October 25, 2022

CGTdAM: Partially supported by CNPq, the Palis Balzan Prize and FAPERJ.
SARI: Partially supported by CNPq, Capes, the Palis Balzan Prize and FAPERJ.

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Abstract

We consider the Lagrange and the Markov dynamical spectra associated with the geodesic flow on surfaces of negative curvature. We show that for a large set of real functions on the unit tangent bundle and typical metrics with negative curvature and finite volume, both the Lagrange and the Markov dynamical spectra have non-empty interiors.

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Mathematics Subject Classification: Primary: 37D40; Secondary: 53C20.

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