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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 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]

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 $$({\mathbb {T}}^2,g)$$ 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]

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]