Finite energy foliations, Floer and contact homology in low dimensions, Hamiltonian dynamics, braids

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A surface of section is a useful device for understanding a dynamical system, a concept introduced by Poincaré in his studies of the 3-body problem. Unfortunately, surfaces of section are generally very difficult to find. Celebrated work of Hofer-Wysocki-Zehnder in the late ¿90¿s related surfaces of section in Hamiltonian systems with objects from the neighbouring field of symplectic geometry, called pseudo-holomorphic curves (which are solutions to a certain partial differential equation and are special minimal surfaces). This opened the possibility of obtaining surfaces of section through a gadget called a finite energy foliation. Recent results by the applicant used these connections to make progress on two questions in ergodic theory, one of these long-standing. Unfortunately, current methods for constructing finite energy foliations crucially lack a certain control related to the asymptotic behaviour of finite energy foliations, which we believe strongly limits applications to dynamics. Our proposal explores a new framework for constructing finite energy foliations with the desirable asymptotic control. The first aim of this proposal is to do this for 2 dimensional Hamiltonian systems. A key new idea will be to take a non-symplectic concept from work of LeCalvez¿ and combine it with an established tool from symplectic geometry called Floer homology. This will result in an interesting new hybrid version of the latter. The work of LeCalvez we refer to was around 2000 where he discovered an object in surface dynamics that shares many features with a finite energy foliation, but remarkably has no direct relation to pseudo-holomorphic curves, or differential equations. We will nevertheless incorporate a concept from his approach into the symplectic category. Besides helping to solve our problem, this should also be of independent interest. The second aim of our proposal is to extend these investigations to a class of 3 dim systems called Reeb flows, which is a natural class of Hamiltonian systems to work with. Here there is no direct analogue of LeCalvez¿ work, but on the symplectic side we will replace Floer homology by contact homology. A third aim is to explore another new variant of Floer and contact homology we propose, which is in some sense ¿transverse" to our investigations above.