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On the graph theorem for Lagrangian invariant tori with totally irrational invariant sets

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Abstract

We show that every \(C^{2}\) Lagrangian invariant torus W of a Tonelli Hamiltonian containing a uniformly continuous curve whose canonical projection has totally irrational homology is a graph, namely, the canonical projection restricted to W is a diffeomorphism. This result extends the graph property obtained by Bangert and Bialy–Polterovich for Lagrangian minimizing tori, invariant by the geodesic flow of a Riemannian metric in the 2-torus, without periodic orbits. Motivated by the famous Hedlund’s examples of Riemannian metrics in the n-torus with n closed, homology independent, minimizing geodesics having minimizing tunnels, we also show that Lagrangian, invariant tori with“large” homology (in the sense of Proposition 4.1) must be graphs. Moreover, we show the \(C^{1}\)-generic nonexistence of Lagrangian invariant tori with“large”homology.

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Authors and Affiliations

  1. ICEX, UFMG, Belo Horizonte, Minas Gerais, Brazil

    Mário J. Dias Carneiro

  2. Dept. de Matematica, PUC, Rio de Janeiro, Brazil

    Rafael O. Ruggiero

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  1. Mário J. Dias Carneiro
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  2. Rafael O. Ruggiero
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Correspondence to Rafael O. Ruggiero.

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Partially supported by PRONEX de Geometria, FAPERJ (Grant No. Pronex/E-26/010.001249/2016)

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Dias Carneiro, M.J., Ruggiero, R.O. On the graph theorem for Lagrangian invariant tori with totally irrational invariant sets. manuscripta math. 171, 423–436 (2023). https://doi.org/10.1007/s00229-022-01391-1

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  • DOI: https://doi.org/10.1007/s00229-022-01391-1

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