Legendrian persistence modules and dynamics

Contact topology and non-equilibrium thermodynamics

“…It is worth mentioning that interlinking is detected in [7] with the help of Legendrian contact homology (LCH) [5], a sophisticated algebraic structure associated to a Legendrian link and involving ideas coming from the string theory. The key ingredient of LCH is a count of special non-compact surfaces in Σ × R whose boundaries lie on Λ 0 × R and Λ 1 × R and which are asymptotic to a collection of the Reeb chords.…”

“…Existence of (robustly) interlinked pairs is a non-trivial phenomenon detected by methods of 'hard' contact topology. We refer to [7,8] for a detailed discussion, as well as to [6] for a related notion in symplectic Hamiltonian dynamics.…”

“…Second, we discuss a mathematical model of non-equilibrium thermodynamics where such trajectories play a role of relaxation processes, and illustrate our results in the case of Glauber dynamics for the Ising model in the mean field approximation. Our starting point is the existence mechanism, called interlinking, for finite time-length trajectories between a pair of Legendrians developed in [7].…”

Exact Lagrangians in four‐dimensional symplectisations

“…In this paper, we produce new examples of closed exact Lagrangians in symplectisations, and also use the same ideas to prove that non‐interlinkedness holds in the dynamical sense defined by [17] by using a similar trick.…”

“…Entov–Polterovich proved the following ‘dynamical interlinkedness’ result in [17, Theorem 1.5]. Let $$\Lambda \subset J^1M \setminus j^10$$ be a closed Legendrian for which there exists a unique transverse Reeb chord from $$j^10$$ to $$\Lambda$$, and such that $$\Lambda$$ has a Chekanov–Eliashberg algebra that admits an augmentation.…”