Locally uniform existence of leafwise fixed points for C0-small Hamiltonian flows & generating systems of symplectic capacities

Abstract

The thesis discusses two topics: existence of leafwise fixed points and generating systems of symplectic capacities. The following results are the main contributions of this thesis: We provide the following solution to the question raised by J. Moser about finding sufficient conditions for the existence of leafwise fixed points:Consider a closed coisotropic submanifold $N_0$ of a symplectic manifold $(M,\omega_0)$. We prove that for every symplectic form $\omega$ that is $C^0$-close to $\omega_0$, every coisotropic submanifold $N$ that is $C^1$-close to $N_0$, and every $\omega$-Hamiltonian flow $(\varphi^t)_{t \in [0,1]}$ the following holds. If the restriction of $\varphi^t$ to $N$ is $C^0$-close to the inclusion $N_0\to M$ for all $t$ and the Hamiltonian vector field is not too big then $\varphi^1$ has a leafwise fixed point with respect to $\omega$ and $N$. This result is optimal in all possible ways in the sense that the conclusion is false if the regularity of any of the three closeness conditions is decreased by 1.We consider the problem by K.~Cieliebak, H.~Hofer, J.~Latschev, and F.~Schlenk (CHLS) that is concerned with finding a minimal generating system for symplectic capacities on a given symplectic category. We show that every countably-generating set of capacities has cardinality bigger than the continuum, provided that the symplectic category contains certain disjoint unions of shells. This appears to be the first result regarding the problem of CHLS, except for a result by D.~McDuff, stating that the ECH-capacities are monotonely generating for the category of ellipsoids in dimension 4. We also prove that every finitely differentiably generating system of symplectic capacities on the category of ellipsoids is uncountable.It implies that the Ekeland-Hofer capacities and the volume capacity do not finitely differentiably generate all generalized capacities on the category of ellipsoids. This answers a variant of a question by CHLS.