Growth and mixing
Krzysztof Frączek Leonid Polterovich
Journal of Modern Dynamics 2008, 2(2): 315-338 doi: 10.3934/jmd.2008.2.315
Given a bi-Lipschitz measure-preserving homeomorphism of a finite dimensional compact metric measure space, consider the sequence of the Lipschitz norms of its iterations. We obtain lower bounds on the growth rate of this sequence assuming that our homeomorphism mixes a Lipschitz function. In particular, we get a universal lower bound which depends on the dimension of the space but not on the rate of mixing. Furthermore, we get a lower bound on the growth rate in the case of rapid mixing. The latter turns out to be sharp: the corresponding example is given by a symbolic dynamical system associated to the Rudin–Shapiro sequence
keywords: Growth rate of homeomorphism the rate of mixing.
Poisson brackets, quasi-states and symplectic integrators
Michael Entov Leonid Polterovich Daniel Rosen
Discrete & Continuous Dynamical Systems - A 2010, 28(4): 1455-1468 doi: 10.3934/dcds.2010.28.1455
This paper is a fusion of a survey and a research article. We focus on certain rigidity phenomena in function spaces associated to a symplectic manifold. Our starting point is a lower bound obtained in an earlier paper with Zapolsky for the uniform norm of the Poisson bracket of a pair of functions in terms of symplectic quasi-states. After a short review of the theory of symplectic quasi-states we extend this bound to the case of iterated Poisson brackets. A new technical ingredient is the use of symplectic integrators. In addition, we discuss some applications to symplectic approximation theory and present a number of open problems.
keywords: symplectic integrators quasi-states Poisson brackets symplectic approximation theory.

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