Positive contactomorphisms and counting chords between Lagrangians

Abstract:

In cooriented contact manifolds there is a natural notion of moving positively, which gives birth to the notion of positive contactomorphisms. The quest of studying their dynamical behaviour includes counting chords between special (Legendrian) submanifolds. Such chords admit a variational principle: they are critical points of an action functional which gives rise to Rabinowitz--Floer homology.

In this talk I will try to avoid the heavy part of the machinery and instead focus on encoding geometric data in such an action functional. In particular I explain how to see that a deformation of the dynamical system preserves the growth of chords. One of the consequences is that exponential growth in the filtered homology implies that every positive contactomorphism has positive topological entropy.