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Birkhoff Program for Geodesic Flows of Surfaces and Applications: Homoclinics

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Abstract

We show that a Kupka–Smale riemannian metric on a closed surface contains a finite primary set of closed geodesics, i.e. they intersect any other geodesic and divide the surface into simply connected regions. From them we obtain a finite set of disjoint surfaces of section of genera 0 or 1, which intersect any orbit of the geodesic flow. As an application we obtain that the geodesic flow of a Kupka–Smale riemannian metric on a closed surface has homoclinic orbits for all branches of all of its hyperbolic closed geodesics.

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Notes

  1. After [12] was published in ArXiv, in [7] appeared another proof that generic 3 dimensional Reeb flows have a Birkhoff section. But the statement of Theorem 1.1 in [7] does not apply to geodesic flows because, for example, the density of closed geodesics in the unit tangent bundle is still not known for generic riemannian metrics.

  2. There are arbitrarily small curves whose image under the Poincaré map have infinite length and large diameter.

  3. A heteroclinic connection is the case in which two components of \(W^s({\dot{\gamma }})\setminus {\dot{\gamma }}\) and \(W^u({\dot{\eta }})\setminus {\dot{\eta }}\) are equal.

  4. This classification is the same as radial and broken binding orbits for broken book decompositions.

  5. Rotating boundary orbits can be hyperbolic or elliptic.

  6. This condition says that the flow rotates more than the surface of section when it approaches its boundary orbit \(\gamma \).

  7. In lemma 3.3 in [12] an argument of Fried is used to show that, since by Proposition 3.8 an orbit \(\kappa \in {K_{fix}}\) has homoclinics in all its branches, one can obtain a Birkhoff section \(\mathcal {B}\) which intersects \(\kappa \) in its interior.

  8. In fact the natural extension of f to a point \(x\in {K_{fix}}\) would be the whole circle of a first intersection of a component of \(W^u(\gamma )\setminus \gamma \), \(\gamma =\psi _{\mathbb R}(x)\), with S. See Fig. 4.

  9. For the boundedness of \(\tau _-\) we apply Proposition 3.12 to the inverse flow \(\psi _{-t}\).

  10. This does not happen on a contact flow but may happen for a reparametrization of the flow.

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Funding

Gonzalo Contreras is partially supported by CONACYT, Mexico, grant A1-S-10145.

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Authors and Affiliations

  1. Centro de Investigación en Matemáticas, Guanajuato, Guanajuato, Mexico

    Gonzalo Contreras

  2. Universidade Federal de Minas Gerais, Belo Horizonte, Minas Gerais, Brazil

    Fernando Oliveira

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  1. Gonzalo Contreras
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  2. Fernando Oliveira
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The contributions of GC happened more in the study of geodesic flows and those of FO in discrete dynamics.

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Correspondence to Fernando Oliveira.

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Gonzalo Contreras is partially supported by CONACYT, Mexico, grant A1-S-10145.

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Contreras, G., Oliveira, F. Birkhoff Program for Geodesic Flows of Surfaces and Applications: Homoclinics. J Dyn Diff Equat (2024). https://doi.org/10.1007/s10884-024-10349-8

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