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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. [ABSTRACT FROM AUTHOR]

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: (1) sharp local systolic inequalities for Riemannian and Finsler metrics close to Zoll ones, (2) the perturbative case of a conjecture of Viterbo on the symplectic capacity of convex bodies, (3) a generalization of Gromov's non-squeezing theorem in the intermediate dimensions for symplectomorphisms that are close to linear ones. [ABSTRACT FROM AUTHOR]

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 invariant 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 weak rescalings. [ABSTRACT FROM AUTHOR]

We study the relationship between a homological capacity c 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 π 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 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 R 3 is proved. [ABSTRACT FROM AUTHOR]

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 Mañé critical value of the universal abelian cover, the Lagrangian system admits a minimal boundary, that is, 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 |$\textrm{T} M$|⁠ , 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 Mañé critical value of the universal abelian cover. This provides in particular a class of nonreversible Finsler metrics on the two-sphere possessing infinitely many closed geodesics. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let α be a contact form on a connected closed three-manifold Σ. The systolic ratio of α is defined as ρsys(α) where Tmin(α) and Vol(α) denote the minimal period of periodic Reeb orbits and the contact volume. The form α is said to be Zoll if its Reeb flow generates a free S1-action on Σ. We prove that the set of Zoll contact forms on Σ locally maximises the systolic ratio in the C3-topology. More precisely, we show that every Zoll form α∗ admits a C3-neighbourhood U in the space of contact forms such that, for every α∈U, there holds ρsys(α)≤ρsys(α∗) with equality if and only if α is Zoll. [ABSTRACT FROM AUTHOR]

We prove that symplectic cohomology for open convex symplectic manifolds is invariant when the symplectic form undergoes deformations which may be nonexact and noncompactly 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. [ABSTRACT FROM AUTHOR]

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 in Calc Var Partial Differ Equ 54(2):1525-1545. doi:, 2015) and at the same time extends the main theorem of Abbondandolo et al. (J Eur Math Soc, , to appear) to the non-exact oscillating case. [ABSTRACT FROM AUTHOR]

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 C2. Our construction generalizes the examples of integrable Finsler metrics on the two-sphere with only two closed geodesics due to Katok. [ABSTRACT FROM AUTHOR]

In this paper we give some positive and negative results about the contact property for the energy levels ${\rm\Sigma}_{c}$ 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. The collected data hint 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 a small energy $c$, there exist a convex hypersurface $N_{c}$ in $\mathbb{C}^{2}$ and a double cover $N_{c}\rightarrow {\rm\Sigma}_{c}$ such that the pull-back of the characteristic distribution on ${\rm\Sigma}_{c}$ is the standard characteristic distribution on $N_{c}$. As a corollary, we prove that there are either two or infinitely many periodic orbits on ${\rm\Sigma}_{c}$. The second alternative holds if there exists a contractible prime periodic orbit. [ABSTRACT FROM PUBLISHER]

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 Mañé critical value of the universal cover there are infinitely many closed magnetic geodesics with energy $$k$$ . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]