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Lagrange and Markov spectra for typical smooth systems

Part of: Dynamical systems with hyperbolic behavior Smooth dynamical systems: general theory

Published online by Cambridge University Press:  23 October 2025

JAMERSON BEZERRA ,
CARLOS GUSTAVO MOREIRA  and
SANDOEL VIEIRA Open the ORCID record for SANDOEL VIEIRA [Opens in a new window]
JAMERSON BEZERRA
Affiliation:
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University , ul. Chopina 12/18, 87-100 Toruń, Poland (e-mail: jdbezerra@mat.umk.pl)
CARLOS GUSTAVO MOREIRA
Affiliation:
Instituto de Matemática Pura e Aplicada (IMPA) , Estrada Dona Castorina, 110, 22460-320 Rio de Janeiro, Brazil (e-mail: gugu@impa.br)
SANDOEL VIEIRA*
Affiliation:
Universidade Federal do Piauí (UFPI) , Rua Dirce Oliveira, 1450, 64048-550 Teresina, Brazil
*
e-mail: sandoel.vieira@ufpi.edu.br
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Abstract

We prove that among the set of pairs ($C^2$-diffeomorphism, $C^1$-potential), there exists a $C^1$-open and dense subset such that either the Lagrange spectrum is finite and the dynamics is a Morse–Smale diffeomorphism or the Lagrange spectrum has positive Hausdorff dimension and the system has positive topological entropy. We also prove that such dichotomy does not hold for typical systems when replacing the Lagrange by the Markov spectrum.

Keywords

Lagrange spectraMarkov spectraHorseshoes in higher dimensionsMorse–Smale

MSC classification

Primary: 37D05: Hyperbolic orbits and sets
Secondary: 37C05: Smooth mappings and diffeomorphisms

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 33
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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