Dressing transformations and Poisson group actions

Group Actions, Linear Transformations, and Flows

It is known that there is a close relationship between matrix groups and linear transformations. The purpose of this exposition is to explore that relationship and to bridge it to the area of applied mathematics. Some known connections between discrete algorithms and differential systems will be used to motivate a larger framework that allows embracing more general matrix groups. Different type...

Poisson-Pinsker factor and infinite measure preserving group actions

We solve the question of the existence of a Poisson-Pinsker factor for conservative ergodic infinite measure preserving action of a countable amenable group by proving the following dichotomy: either it has totally positive Poisson entropy (and is of zero type), or it possesses a Poisson-Pinsker factor. If G is abelian and the entropy positive, the spectrum is absolutely continuous (Lebesgue co...

Integrability and Reduction of Poisson Group Actions 1

In this paper we study Poisson actions of complete Poisson groups, without any connectivity assumption or requiring the existence of a momentum map. For any complete Poisson group G with dual G⋆ we obtain a suitably connected integrating symplectic double groupoid S. As a consequence, the cotangent lift of a Poisson action on an integrable Poisson manifold P can be integrated to a Poisson actio...

Deck Transformations and Group Actions 70. Additional Topics

On Poisson Nonlinear Transformations

We construct the family of Poisson nonlinear transformations defined on the countable sample space of nonnegative integers and investigate their trajectory behavior. We have proved that these nonlinear transformations are regular.