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We prove that all normalized symplectic capacities coincide on smooth domains in $\mathbb C^n$ which are $C^2$-close to the Euclidean ball, whereas this fails for some smooth domains which are just $C^1$-close to the ball. We also prove that all symplectic capacities whose value on ellipsoids agrees with that of the $n$-th Ekeland-Hofer capacity coincide in a $C^2$-neighborhood of the Euclidean ball of $\mathbb C^n$. These results are deduced from a general theorem about contact forms which are $C^2$-close to Zoll ones, saying that these contact forms can be pulled back to suitable "quasi-invariant" contact forms. We relate all this to the question of the existence of minimizing geodesics in the space of contact forms equipped with a Banach-Mazur pseudo-metric. Using some new spectral invariants for contact forms, we prove the existence of minimizing geodesics from a Zoll contact form to any contact form which is $C^2$-close to it. This paper also contains an appendix in which we review the construction of exotic ellipsoids by the Anosov-Katok conjugation method, as these are related to the above mentioned pseudo-metric., Comment: 70 pages, comments welcome

We prove a normal form for strong magnetic fields on a closed, oriented surface and use it to derive two dynamical results for the associated flow. First, we show the existence of KAM tori and trapping regions provided a natural non-resonance condition holds. Second, we prove that the flow cannot be Zoll unless (i) the Riemannian metric has constant curvature and the magnetic function is constant, or (ii) the magnetic function vanishes and the metric is Zoll. We complement the second result by exhibiting an exotic magnetic field on a flat two-torus yielding a Zoll flow for arbitrarily small rescalings., Comment: 19 pages, final version to appear in ETDS, several typos corrected, funding acknowledgments updated. Comments are very welcome!

In this paper we study some aspects of integrable magnetic systems on the two-torus. On the one hand, we construct the first non-trivial examples with the property that all magnetic geodesics with unit speed are closed. On the other hand, we show that those integrable magnetic systems admitting a global surface of section satisfy a sharp systolic inequality., Comment: 11 pages, 2 figures

Let $\Sigma$ be a connected closed three-manifold, and let $t_\Sigma$ be the order of the torsion subgroup of $H_1(\Sigma;\mathbb Z)$. For a contact form $\alpha$ on $\Sigma$, we denote by $\mathrm{Volume}(\alpha)$ the contact volume of $\alpha$, and by $T_{\min}(\alpha)$ and $T_{\max}(\alpha)$ the minimal period and the maximal period of prime periodic orbits of the Reeb flow of $\alpha$ respectively. We say that $\alpha$ is Zoll if its Reeb flow generates a free $S^1$-action on $\Sigma$. We prove that every Zoll contact form $\alpha_*$ on $\Sigma$ admits a $C^3$-neighbourhood $\mathcal U$ in the space of contact forms such that \[ t_\Sigma T_{\min}(\alpha)^2\leq \mathrm{Volume}(\alpha)\leq t_\Sigma T_{\max}(\alpha)^2,\qquad \forall\,\alpha\in\mathcal U, \] and any of the equalities holds if and only if $\alpha$ is Zoll. We extend the above picture to odd-symplectic forms $\Omega$ on $\Sigma$ of arbitrary odd dimension. We define the volume of $\Omega$, which generalises both the contact volume and the Calabi invariant of Hamiltonian functions, and the action of closed characteristics of $\Omega$, which generalises both the period of periodic Reeb orbits and the action of fixed points of Hamiltonian diffeomorphisms. We say that $\Omega$ is Zoll if its characteristics are the orbits of a free $S^1$-action on $\Sigma$. We prove that the volume and the action of a Zoll odd-symplectic form satisfy a certain polynomial equation. This builds the equality case of a conjectural local systolic-diastolic inequality for odd-symplectic forms, which we establish in some cases. This inequality recovers the inequality between the minimal action and the Calabi invariant of Hamiltonian isotopies $C^1$-close to the identity on a closed symplectic manifold, as well as the local contact systolic-diastolic inequality above. Finally, applications to magnetic geodesics are discussed., Comment: The contents of the article is now divided into three articles available at arXiv:1902.01249, arXiv:1902.01261, and arXiv:1902.01262

We prove several new results concerning action minimizing periodic orbits of Tonelli Lagrangian systems on an oriented closed surface $M$. More specifically, we show that for every energy larger than the maximal energy of a constant orbit and smaller than or equal to the Ma\~n\'e critical value of the universal abelian cover, the Lagrangian system admits a minimal boundary, i.e. a global minimizer of the Lagrangian action on the space of smooth boundaries of open sets of $M$. We also extend the celebrated graph theorem of Mather in this context: in the tangent bundle $TM$, the union of the supports of all lifted minimal boundaries with a given energy projects injectively to the base $M$. Finally, we prove the existence of action minimizing simple periodic orbits on energies just above the Ma\~n\'e critical value of the universal abelian cover. This provides in particular a class of non-reversible Finsler metrics on the 2-sphere possessing infinitely many closed geodesics., Comment: 31 pages, 4 figures. This version also incorporates the results of arXiv:1702.08815 (the preprint arXiv:1702.08815 has been withdrawn from the arXiv, and will not be submitted for publication)

We introduce a new critical value $c_\infty(L)$ for Tonelli Lagrangians $L$ on the tangent bundle of the 2-sphere without minimizing measures supported on a point. We show that $c_\infty(L)$ is strictly larger than the Ma\~n\'e critical value $c(L)$, and on every energy level $e\in(c(L),c_\infty(L))$ there exist infinitely many periodic orbits of the Lagrangian system of $L$, one of which is a local minimizer of the free-period action functional. This has applications to Finsler metrics of Randers type on the 2-sphere. We show that, under a suitable criticality assumption on a given Randers metric, after rescaling its magnetic part with a sufficiently large multiplicative constant, the new metric admits infinitely many closed geodesics, one of which is a waist. Examples of critical Randers metrics include the celebrated Katok metric., Comment: The results in this preprint are now incorporated in our work arXiv:1705.02488 with Luca Asselle. This preprint will not be submitted for publication

We consider magnetic Tonelli Hamiltonian systems on the cotangent bundle of the 2-sphere, where the magnetic form is not necessarily exact. It is known that, on very low and on high energy levels, these systems may have only finitely many periodic orbits. Our main result asserts that almost all energy levels in a precisely characterized intermediate range $(e_0,e_1)$ possess infinitely many periodic orbits. Such a range of energies is non-empty, for instance, in the physically relevant case where the Tonelli Lagrangian is a kinetic energy and the magnetic form is oscillating (in which case, $e_0=0$ is the minimal energy of the system)., Comment: 17 pages, 1 figure

These notes were prepared in occasion of a mini-course given by the author at the "CIMPA Research School - Hamiltonian and Lagrangian Dynamics" (10-19 March 2015 - Salto, Uruguay). The talks were meant as an introduction to the problem of finding periodic orbits of prescribed energy for autonomous Tonelli Lagrangian systems on the twisted cotangent bundle of a closed manifold. In the first part of the lecture notes, we put together in a general theorem old and new results on the subject. In the second part, we focus on an important class of examples: magnetic flows on surfaces. For such systems, we discuss a special method, originally due to Taimanov, to find periodic orbits with low energy and we study in detail the stability properties of the energy levels., Comment: One figure. To appear in a special issue of "Publicaciones Matem\'aticas del Uruguay" dedicated to Ricardo Ma\~n\'e. Comments are welcome

Let $(\mathbb T^2,g)$ be a Riemannian two-torus and let $\sigma$ be an oscillating $2$-form on $\mathbb T^2$. We show that for almost every small positive number $k$ the magnetic flow of the pair $(g,\sigma)$ has infinitely many periodic orbits with energy $k$. This result complements the analogous statement for closed surfaces of genus at least $2$ [Asselle and Benedetti, Calc. Var. Partial Differential Equations, 2015] and at the same time extends the main theorem in [Abbondandolo, Macarini, Mazzucchelli, and Paternain, J. Eur. Math. Soc. (JEMS), to appear] to the non-exact oscillating case., Comment: 15 pages. Revised version incorporating the precious comments of the referee and the notion of essential family suggested to us by M. Mazzucchelli. Comments are very welcome. To appear in Mathematische Zeitschrift

We show that there exist non-exact magnetic flows on the two-sphere having an energy level whose double cover is strictly contactomorphic to an irrational ellipsoid in $\mathbb C^2$. Our construction generalizes the examples of integrable Finsler metrics on the two-sphere with only two closed geodesics due to A. Katok., Comment: 13 pages. Second version. We have added a sharper example in the larger class of Lagrangians attaining a Morse-Bott non-degenerate minimum at the zero section. Comments, questions, and suggestions are very welcome. To appear in the Bulletin of the London Mathematical Society

We prove that there exist periodic orbits on almost all compact regular energy levels of a Hamiltonian function defined on a twisted cotangent bundle over the two-sphere. As a corollary, given any Riemannian two-sphere and a magnetic field on it, there exists a closed magnetic geodesic for almost all kinetic energy levels., Comment: New title and abstract; same content; to appear in J. Mod. Dyn

Let $M$ be a closed manifold and consider the Hamiltonian flow associated to an autonomous Tonelli Hamiltonian $H:T^*M\rightarrow \mathbb R$ and a twisted symplectic form. In this paper we study the existence of contractible periodic orbits for such a flow. Our main result asserts that if $M$ is not aspherical, then contractible periodic orbits exist for almost all energies above the maximum critical value of $H$., Comment: 21 pages. We have generalized the results of the previous version to a larger class of manifolds and of energy values. Remarks and comments are very welcome. To appear on Journal of Topology and Analysis

In this paper we consider oscillating non-exact magnetic fields on surfaces with genus at least two and show that for almost every energy level $k$ below a certain value $\tau_+^*(g,\sigma)$ less than or equal to the "Ma\~n\'e critical value of the universal cover" there are infinitely many closed magnetic geodesics with energy $k$., Comment: In this version we corrected some minor inaccuracies and we improved the exposition following the precious suggestions of the anonymous referee. Accepted for publication in "Calculus of Variations and Partial Differential Equations". Comments are very welcome. arXiv admin note: text overlap with arXiv:1404.7641 by other authors

In this paper we give some positive and negative results about the contact property for the energy levels $\Sigma_c$ of a symplectic magnetic field on $S^2$. In the first part we focus on the case of the area form on a surface of revolution. We state a sufficient condition for an energy level to be of contact type and give an example where the contact property fails. If the magnetic curvature is positive, the dynamics and the action of invariant measures can be numerically computed. This hints at the conjecture that an energy level of a symplectic magnetic field with positive magnetic curvature should be of contact type. In the second part we show that, for small energies, there exists a convex hypersurface $N_c$ in $\mathbb C^2$ and a covering map $p_c:N_c \rightarrow \Sigma_c$ such that the pull-back via $p_c$ of the characteristic distribution on $\Sigma_c$ is the standard characteristic distribution on $N_c$. As a corollary we prove that there are either two or infinitely many periodic orbits on $\Sigma_c$. The second alternative holds if there exists a contractible prime periodic orbit., Comment: 31 pages, 1 figure. Minor typos corrected. Published online in "Ergodic Theory and Dynamical Systems"