Quasi-morphisms and quasi-states in symplectic topology

Abstract

We discuss certain "almost homomorphisms" and "almost linear" functionals that have appeared in symplectic topology and their applications concerning Hamiltonian dynamics, functional-theoretic properties of Poisson brackets and algebraic and metric properties of symplectomorphism groups.

A characterization of heaviness in terms of relative symplectic cohomology

“…This implication was communicated to us by Adi Dickstein, Yaniv Ganor, Leonid Polterovich and Frol Zapolsky after the first preprint version of the paper was released. It would take us too far to try to introduce the relevant definitions or notions regarding symplectic quasi-states here, so we refer the reader to [5] for those. As noted in Remark 1.2, the results stated in the introduction are, in fact, proved for all idempotents (not just for the unit) and we use this below.…”

“…Corollary 1.10. Every symplectic quasi-state constructed in [5,Theorem 3.1] corresponding to the unit in a field factor of 𝑄𝐻(𝑀; Λ) is dispersion-free.…”

“…Consider 𝑓 as an involutive map 𝑓 ∶ 𝑀 → ℝ. Corollary 1.9 implies that this map has a heavy fiber, say at the point 𝑡 ∈ ℝ. Since 𝑧 is a genuine quasi-state, this fiber is also superheavy by [5,Theorem 4.1.e]. It then follows that the pushforward functional 𝑓 * 𝑧 is, in fact, the evaluation at 𝑡 directly from the definition of heavy and superheavy (see, e.g., Definition 2.8).…”

Towards a dynamical interpretation of Hamiltonian spectral invariants on surfaces

“…for any autonomous function H. The functional ζ was introduced by Entov and Polterovich in [8] and in their terminology it is referred to as a (partial) symplectic quasi-state. Partial and genuine symplectic quasi states have been constructed on a large class of symplectic manifolds; see [6] for a survey of the subject. It is well-known that the quasi-state on S 2 admits a very simple description [7].…”

A Class of New Bi-Invariant Metrics on the Hamiltonian Diffeomorphism Groups

“…In 1989, H. Hofer [12] constructed a remarkable bi-invariant Finsler metric on the group of compactly supported Hamiltonian diffeomorphisms Ham(M, ω) of a symplectic manifold (M, ω), nowadays known as Hofer metric. Since then the intrinsic geometry of it has been being a very active and fruitful research field in symplectic topology and Hamiltonian dynamics (see the books [14,18,27], and the surveys [10,19,28,24] and references therein for current progress situation).…”