Franks' lemma for $\mathbf{C}^2$-Mañé perturbations of Riemannian metrics and applications to persistence

Ayadi  Lazrag; Ludovic  Rifford; Rafael O.  Ruggiero

doi:10.3934/jmd.2016.10.379

Article Contents

2016, Volume 10: 379-411. Doi: 10.3934/jmd.2016.10.379

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Franks' lemma for $\mathbf{C}^2$-Mañé perturbations of Riemannian metrics and applications to persistence

1.
Université Nice Sophia Antipolis, CNRS, LJAD, UMR, 7351, 06100 Nice
2.
Université Nice Sophia Antipolis, Institut Universitaire de France, CNRS, LJAD, UMR 7351, 06100 Nice
3.
PUC-Rio, Departamento de Matemática, Rua Marqués de São Vicente 225, Gávea, 22450-150, Rio de Janeiro

Received: March 31, 2014

Revised: April 30, 2016

Published: August 2016

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Abstract

We prove a uniform Franks' lemma at second order for geodesic flows on a compact Riemannian manifold and apply the result in persistence theory. Our approach, which relies on techniques from geometric control theory, allows us to show that Mañé (i.e., conformal) perturbations of the metric are sufficient to achieve the result.

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Mathematics Subject Classification: Primary: 93C05; Secondary: 37C20, 70G45, 70H14.

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