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Minimal number of periodic orbits for non-singular Morse–Smale flows in odd dimension

Volume 267 / 2024

Maria Alice Bertolim, Christian Bonatti, Margarida Pinheiro Mello, Gioia Maria Vago Fundamenta Mathematicae 267 (2024), 25-85 MSC: Primary 37B30; Secondary 37C27, 37D15 DOI: 10.4064/fm231019-5-7 Published online: 9 October 2024

Abstract

We consider the couples $(M, \Phi )$ where $M$ is an odd-dimensional compact manifold with boundary, endowed with a non-singular Morse–Smale flow $\Phi $, satisfying some given homological boundary information. We compute, in terms of that information, a number $p_{\rm min}$ such that any non-singular Morse–Smale flow $\Phi $ on any manifold $M$ satisfying the given abstract homological data must have at least $p_{\rm min}$ closed periodic orbits.

Moreover, we provide, for any initial homological data, a non-singular Morse–Smale model $(M_0, \Phi _0)$ for which $p_{\rm min}$ is attained. In the general case of a couple $(M, \Phi )$ satisfying the given homological information, such a number $p_{\rm min}$ is a lower bound.

The algorithm underlying this computation is based on optimization theory in network flows and transport systems.

Authors

  • Maria Alice BertolimInstitut de Mathématiques de Bourgogne
    UMR 5584, CNRS
    ESEO – Grande École d’Ingénieurs
    F-21000 Dijon, France
    e-mail
  • Christian BonattiInstitut de Mathématiques de Bourgogne
    UMR 5584, CNRS
    Université de Bourgogne
    F-21000 Dijon, France
    e-mail
  • Margarida Pinheiro MelloIMECC/UNICAMP
    Universidade de Campinas
    13083-970 Campinas, Brazil
    e-mail
  • Gioia Maria VagoInstitut de Mathématiques de Bourgogne
    UMR 5584, CNRS
    Université de Bourgogne
    F-21000 Dijon, France
    e-mail

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