Journal of Modern Dynamics (JMD)

Quasi-states and the Poisson bracket on surfaces

Pages: 465 - 475, Issue 3, July 2007      doi:10.3934/jmd.2007.1.465

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Frol Zapolsky - School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel (email)

Abstract: We present a convexity-type result concerning simple quasi-states on closed manifolds. As a corollary, an inequality emerges which relates the Poisson bracket to the measure of non-additivity of a simple quasi-state on a closed surface equipped with an area form. In addition, we prove that the uniform norm of the Poisson bracket of two functions on a surface is stable from below under $C^0$-perturbations.

Keywords:  quasi-states, Poisson bracket, surfaces.
Mathematics Subject Classification:   Primary: 53D99 Secondary: 28C15.

Received: February 2007;      Revised: March 2007;      Available Online: April 2007.