Journal of Modern Dynamics (JMD)

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

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

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Ayadi Lazrag - Université Nice Sophia Antipolis, CNRS, LJAD, UMR, 7351, 06100 Nice, France (email)
Ludovic Rifford - Université Nice Sophia Antipolis, Institut Universitaire de France, CNRS, LJAD, UMR 7351, 06100 Nice, France (email)
Rafael O. Ruggiero - PUC-Rio, Departamento de Matemática, Rua Marqués de São Vicente 225, Gávea, 22450-150, Rio de Janeiro, Brazil (email)

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.

Keywords:  Franks’ Lemma, geodesic flows, Mañé’s genericity, control theory of linear systems.
Mathematics Subject Classification:  Primary: 93C05; Secondary: 37C20, 70G45, 70H14.

Received: April 2014;      Revised: May 2016;      Available Online: September 2016.