Quasi-morphisms and the Poisson bracket

By Michael Entov, L. Polterovich, F. Zapolsky. 2006

69 citations recorded by the source corpus.

For a class of symplectic manifolds, we introduce a functional which assigns a real number to any pair of continuous functions on the manifold. This functional has a number of interesting properties. On the one hand, it is Lipschitz with respect to the uniform norm. On the other hand, it serves as a measure of non-commutativity of functions in the sense of the Poisson bracket, the operation which involves first derivatives of the functions. Furthermore, the same functional gives rise to a non-trivial lower bound for the error of the simultaneous measurement of a pair of non-commuting Hamiltonians. These results manifest a link between the algebraic structure of the group of Hamiltonian diffeomorphisms and the function theory on a symplectic manifold. The above-mentioned functional comes from a special homogeneous quasi-morphism on the universal cover of the group, which is rooted in the Floer theory.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/77ac9a3a862b65cd75e910e06a362ce439b7c144

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms (2010, background): “M; !/ and to rigidity properties of subsets of M ; see for instance Entov and Polterovich [12; 15] and Entov, Polterovich and Zapolsky [16].”
Commuting Hamiltonians and Hamilton-Jacobi multi-time equations (2005, methodology): “j i is closed3. We refer to [Humili`ere 2] for an approach along these lines of the GromovEliashberg theorem. We also refer to improvemets of the above result from a quantitative point of view due to [Entov-Polterovich-Zapolsky] using quasi-states. This has been extended by V. Humili`ere to other re”
Partial quasi-morphisms and quasi-states on cotangent bundles, and symplectic homogenization (2011, methodology): “. Point (i) is proved by invoking the semi-homogeneity of µa to obtain the desired identity ﬁrst for natural, then rational, and ﬁnally, using continuity, for arbitrary λ≥0. Point (v) is proved as in [EPZ], carefully keeping track of the constants. 3.2 Equivalence to symplectic homogenization on Tn ”