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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 show that on the unit disc cotangent bundle of flat Riemannian tori, all normalized capacities coincide with twice the systole. The same result holds for flat, reversible Finsler tori and normalized capacities that are greater than or equal to the Hofer-Zehnder capacity., Comment: 6 pages. Comments are welcome! Fixed some typos. To appear in Proceedings of the AMS

We construct an infinite-dimensional family of smooth integrable magnetic systems on the two-torus which are Zoll, meaning that all the unit-speed magnetic geodesics are periodic. The metric and the magnetic field of such systems are arbitrarily close to the flat metric and to a given constant magnetic field. This extends to the magnetic setting a famous result by Guillemin on the two-sphere. We characterize Zoll magnetic systems as zeros of a suitable action functional $S$, and then look for its zeros by means of a Nash-Moser implicit function theorem. This requires showing the right-invertibility of the linearized operator $\mathrm{d} S$ in a neighborhood of the flat metric and constant magnetic field, and establishing tame estimates for the right inverse. As key step we prove the invertibility of the normal operator $\mathrm{d} S\circ \mathrm{d} S^*$ which, unlike in Guillemin's case, is pseudo-differential only at the highest order. We overcome this difficulty noting that, by the asymptotic properties of Bessel functions, the lower order expansion of $\mathrm{d} S \circ \mathrm{d}S^*$ is a sum of Fourier integral operators. We then use a resolvent identity decomposition which reduces the problem to the invertibility of $\mathrm{d} S \circ \mathrm{d} S^*$ restricted to the subspace of functions corresponding to high Fourier modes. The inversion of such a restricted operator is finally achieved by making the crucial observation that lower order Fourier integral operators satisfy asymmetric tame estimates., Comment: 25 pages; added references and improved exposition in the introduction; fixed minor inaccuracies following referee's report

In these notes we discuss Lorentz-Finsler metrics, a notion originated in relativity theory, on certain groups of symplectic and contact transformations. Some basic geometric questions arising in this context concerning distance, geodesics and their conjugate points, and existence of a time function, turn out to be related to a variety of subjects including the contact systolic problem, group quasi-morphisms, the Monge-Amp\`ere equation, and a subtle interplay between symplectic rigidity and flexibility. We discuss these interrelations, providing necessary preliminaries, and formulate a number of open questions., Comment: 103 pages, 2 figures

Our aim is to give a friendly introduction for students to systolic inequalities. We will stress the relationships between the classical formulation for Riemannian metrics and more recent developments related to symplectic measurements and the Viterbo conjecture. This will give us a perfect excuse to introduce the reader to some important ideas in Riemannian and symplectic geometry., Comment: acknowledgements updated, 31 pages, 9 figures, these notes are an expanded version of two talks given at the Dutsch Differential Topology and Geometry Seminar on November 27, 2020

We study the relationship between a homological capacity $c_{\mathrm{SH}^+}(W)$ for Liouville domains $W$ defined using positive symplectic homology and the existence of periodic orbits for Hamiltonian systems on $W$: If the positive symplectic homology of $W$ is non-zero, then the capacity yields a finite upper bound to the $\pi_1$-sensitive Hofer-Zehnder capacity of $W$ relative to its skeleton and a certain class of Hamiltonian diffeomorphisms of $W$ has infinitely many non-trivial contractible periodic points. En passant, we give an upper bound for the spectral capacity of $W$ in terms of the homological capacity $c_{\mathrm{SH}}(W)$ defined using the full symplectic homology. Applications of these statements to cotangent bundles are discussed and use a result by Abbondandolo and Mazzucchelli in the appendix, where the monotonicity of systoles of convex Riemannian two-spheres in $\mathbb R^3$ is proved., Comment: 30 pages, 1 figure, appendix by Alberto Abbondandolo and Marco Mazzucchelli, comments welcome! Corrected an inconsistency in our convention pointed out by Pierre-Alexandre Mailhot and Egor Shelukhin

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!

We prove a normal form for contact forms close to a Zoll one and deduce that Zoll contact forms on any closed manifold are local maximizers of the systolic ratio. Corollaries of this result are: (i) sharp local systolic inequalities for Riemannian and Finsler metrics close to Zoll ones, (ii) the perturbative case of a conjecture of Viterbo on the symplectic capacity of convex bodies, (iii) a generalization of Gromov's non-squeezing theorem in the intermediate dimensions for symplectomorphisms that are close to linear ones., Comment: 63 pages; v3: perturbative version of the Viterbo conjecture now proven for arbitrary symplectic capacities, added more properties to the normal form, added a statement on the local rigidity of Zoll contact forms

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

The aim of this paper is to formulate a local systolic inequality for odd-symplectic forms (also known as Hamiltonian structures) and to establish it in some basic cases. Let $\Omega$ be an odd-symplectic form on an oriented closed manifold $\Sigma$ of odd dimension. We say that $\Omega$ is Zoll if the trajectories of the flow given by $\Omega$ are the orbits of a free $S^1$-action. After defining the volume of $\Omega$ and the action of its periodic orbits, we prove that the volume and the action satisfy a polynomial equation, provided $\Omega$ is Zoll. This builds the equality case of a conjectural systolic inequality for odd-symplectic forms close to a Zoll one. We prove the conjecture when the $S^1$-action yields a flat $S^1$-bundle or $\Omega$ is quasi-autonomous. In particular the conjecture is established in dimension three. This new inequality recovers the contact systolic inequality as well as the inequality between the minimal action and the Calabi invariant for Hamiltonian isotopies $C^1$-close to the identity on a closed symplectic manifold. Applications to the study of periodic magnetic geodesics on closed orientable surfaces is given in the companion paper available at arXiv:1902.01262., Comment: 52 pages, a revised version of Part II in "A local systolic-diastolic inequality in contact and symplectic geometry" arXiv:1801.00539 (now withdrawn)

We apply a local systolic-diastolic inequality for contact forms and odd-symplectic forms on three-manifolds to bound the magnetic length of closed curves with prescribed geodesic curvature (also known as magnetic geodesics) on an oriented closed surface. Our results hold when the prescribed curvature is either close to a Zoll one or large enough., Comment: 26 pages, a revised version of Part III in "A local systolic-diastolic inequality in contact and symplectic geometry" arXiv:1801.00539 (now withdrawn)

Let $\alpha$ be a contact form on a connected closed three-manifold $\Sigma$. The systolic ratio of $\alpha$ is defined as $\rho_{\mathrm{sys}}(\alpha):=\tfrac{1}{\mathrm{Vol}(\alpha)}T_{\min}(\alpha)^2$, where $T_{\min}(\alpha)$ and $\mathrm{Vol}(\alpha)$ denote the minimal period of periodic Reeb orbits and the contact volume. The form $\alpha$ is said to be Zoll if its Reeb flow generates a free $S^1$-action on $\Sigma$. We prove that the set of Zoll contact forms on $\Sigma$ locally maximises the systolic ratio in the $C^3$-topology. More precisely, we show that every Zoll form $\alpha_*$ admits a $C^3$-neighbourhood $\mathcal U$ in the space of contact forms such that, for every $\alpha\in\mathcal U$, there holds $\rho_{\mathrm{sys}}(\alpha)\leq \rho_{\mathrm{sys}}(\alpha_*)$ with equality if and only if $\alpha$ is Zoll., Comment: 42 pages, a revised version of Part I in "A local systolic-diastolic inequality in contact and symplectic geometry" arXiv:1801.00539 (now withdrawn), accepted for publication in JEMS

We prove that symplectic cohomology for open convex symplectic manifolds is invariant when the symplectic form undergoes deformations which may be non-exact and non-compactly supported, provided one uses the correct local system of coefficients in Floer theory. As a sample application beyond the Liouville setup, we describe in detail the symplectic cohomology for disc bundles in the twisted cotangent bundle of surfaces, and we deduce existence results for periodic magnetic geodesics on surfaces. In particular, we show the existence of geometrically distinct orbits by exploiting properties of the BV-operator on symplectic cohomology., Comment: 48 pages, 1 figure. Results are unchanged. Additional details were added in the Introduction and in the proof of Prop.3.9. To appear in: International Journal of Mathematics

This is the author's PhD Thesis (University of Cambridge, 2014) in its original form. In the first part, using an invariance result, we compute the symplectic homology of contact-type energy levels for magnetic systems on surfaces, provided the energy is very large or very small. In the second part, which is partially contained in the later paper (Benedetti, Ergod. Theory Dynam. Syst., 2016), we discuss some rotationally symmetric examples and establish dynamical convexity for symplectic magnetic flows on low energy levels., Comment: 132 pages. The proof of Proposition 4.20 contains a gap, but the statement of that proposition is still true

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