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Gonzalo Contreras

Publications and source records attributed to Gonzalo Contreras.

At least 19 recordsLinked to original sources
arXiv2025

Closed geodesics and the first Betti number

We prove that, on any closed manifold of dimension at least two with non-trivial first Betti number, a $C^\infty$ generic Riemannian metric has infinitely many closed geodesics, and indeed closed geodesics of arbitrarily large length. We derive this existence result combining a theorem of Mañé together with the following new theorem of independent interest: the existence of minimal closed geodesics, in the sense of Aubry-Mather theory, implies the existence of a transverse homoclinic, and thus of a horseshoe, for the geodesic flow of a suitable $C^\infty$-close Riemannian metric.

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arXiv2024

Minimization and Hyperbolicity

In this paper we study the relationship between the strict locally minimizing orbits for time dependent lagrangian systems and hyperbolicity properties of the corresponding lagrangian flow.

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arXiv2024

Surfaces of section for geodesic flows of closed surfaces

We prove several results concerning the existence of surfaces of section for the geodesic flows of closed orientable Riemannian surfaces. The surfaces of section $Σ$ that we construct are either Birkhoff sections, meaning that they intersect every sufficiently long orbit segment of the geodesic flow, or at least they have some hyperbolic components in $\partialΣ$ as limit sets of the orbits of the geodesic flow that do not return to $Σ$. In order to prove these theorems, we provide a study of configurations of simple closed geodesics of closed orientable Riemannian surfaces, which may have independent interest. Our arguments are based on the curve shortening flow.

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arXiv2024

Proof of the $C^2$ Mañé's conjecture on surfaces

We prove that $C^2$ generic hyperbolic Mañé sets contain a periodic periodic orbit. In dimension 2, adding a result by Contreras, Figalli, Rifford, which states that $C^2$ generic Mañé sets are hyperbolic; we obtain Mañé's Conjecture for surfaces in the $C^2$ topology: Given a Tonelli Lagrangian $L$ on a compact surface $M$ there is a $C^2$ open and dense set of functions $f:M\to\mathbb{R}$ such that the Mañé set of the Lagrangian $L+f$ is a hyperbolic periodic orbit.

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arXiv2024

Proof of the $C^2$-stability conjecture for geodesic flows of closed surfaces

We prove that a $C^2$-generic Riemannian metric on a closed surface has either an elliptic closed geodesic or an Anosov geodesic flow. As a consequence, we prove the $C^2$-stability conjecture for Riemannian geodesic flows of closed surfaces: a $C^2$-structurally stable Riemannian geodesic flow of a closed surface is Anosov. In order to prove these statements, we establish a general result that may be of independent interest and provides sufficient conditions for a Reeb flow of a closed 3-manifold to be Anosov.

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arXiv2024

Positive topological entropy of Tonelli Lagrangian flows

We study the topological entropy of the Lagrangian flow restricted to an energy level $E_{L}^{-1}(c) \subset TM$ for $ c >e_0(L)$. We prove that if the flow of the Tonelli Lagrangian $ L: M \to \mathbb{R}$, on a closed manifold of dimension $ n+1$, has a non-hyperbolic closed orbit or an infinite number of closed orbits with energy $ c>e_0(L)$ and satisfies certain open dense conditions, then there exist a smooth potential $ u: M\to \mathbb{R} $, with $ C^2$-norm arbitrarily small, such that the flow of the perturbed Lagrangian $ L_u=L-u$ restricted to $E_{L_u}^{-1}(c)$ has positive topological entropy. The proof of this result is based on an analog version of the Franks' Lemma for Lagrangian flows and Mañé's techniques on dominated splitting. As an application, we show that if $\dim (M)=2$ and $c > e_0(L)$, then $ L$ admits a $C^2$-perturbation by a smooth potential $u$, such that, the perturbed flow $ϕ_t^{L_u}\big{|}_{E_{L_u}^{-1}(c)}$ has positive topological entropy.

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arXiv2023

The ideal boundary and the accumulation lemma

Let $S$ be a connected surface possibly with boundary, $μ$ a finite Borel measure which is positive on open sets and $f:S\to S$ a homeomorphism preserving $μ$. We prove that if $K$ is a compact connected subset of $S$ and $L$ is a branch of a hyperbolic periodic point if $f$ then $L\cap K\ne\emptyset$ implies $L\subset K$. This is called the accumulation lemma. For this we develop a classification of connected surfaces with boundary and a characterization of residual domains of compact subsets with finitely many connected components in a connected surface with boundary.

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arXiv2023

No elliptic points from fixed prime ends

We consider area preserving maps of surfaces and extend Mather's result on the equality of the closure of the four branches of saddles. He assumed elliptic fixed points to be Moser stable, while we require only that the derivative at this points to be a rotation by an angle different from zero. There are many results in the literature which require the hypothesis that elliptic periodic points be Moser stable that now can be extended to the case that the derivative at these points be an irrational rotation. The key point is to give more information on Cartwright and Littlewood's fixed point theorem, to show that the fixed point obtained by a fixed prime end can not be elliptic. Hypotheses then became easier to verify: non degeneracy of fixed points and nonexistence of saddle connections. As an application we show that the result immediately implies that for the standard map family, for all values of the parameter, except one, the principal hyperbolic fixed point has homoclinic points. We also extend results to surfaces with boundary in order to be applicable to return maps to surfaces of section and broken book decompositions.

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arXiv2021

Existence of Birkhoff sections for Kupka-Smale Reeb flows of closed contact 3-manifolds

A Reeb vector field satisfies the Kupka-Smale condition when all its closed orbits are non-degenerate, and the stable and unstable manifolds of its hyperbolic closed orbits intersect transversely. We show that, on a closed 3-manifold, any Reeb vector field satisfying the Kupka-Smale condition admits a Birkhoff section. In particular, this implies that the Reeb vector field of a $C^\infty$-generic contact form on a closed 3-manifold admits a Birkhoff section, and that the geodesic vector field of a $C^\infty$-generic Riemannian metric on a closed surface admits a Birkhoff section.

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arXiv2021

Generic Mañé sets

We prove that $C^2$ generic hyperbolic Mañé sets contain a periodic orbit. In dimesion 2, adding a result with A. Figalli and L. Rifford, we obtain Mañé's Conjecture for surfaces in the $C^2$ topology.

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arXiv2018

On finite quotient Aubry set for generic geodesic flows

We study the structure of the Mather and Aubry sets for the family of lagrangians given by the kinetic energy associated to a riemannian metric $ g$ on a closed manifold $ M$. In this case the Euler-Lagrange flow is the geodesic flow of $(M,g)$. We prove that there exists a residual subset $ \mathcal G$ of the set of all conformal metrics to $g$, such that, if $ \overline g \in \mathcal G$ then the corresponding geodesic flow has a finitely many ergodic c-minimizing measures, for each non-trivial cohomology class $ c \in H^1(M,\mathbb{R})$. This implies that, for any $ c \in H^1(M,\mathbb{R})$, the quotient Aubry set for the cohomology class c has a finite number of elements for this particular family of lagrangian systems.

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arXiv2014

Homogenization on manifolds

We present a theorem by Contreras, Iturriaga and Siconolfi in which we give a setting to generalize the homogenization of the Hamilton-Jacobi equation from tori to other manifolds.

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arXiv2014

Homogenization on arbitrary manifolds

We describe a setting for homogenization of convex hamiltonians on abelian covers of any compact manifold. In this context we also provide a simple variational proof of standard homogenization results.

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