I’m so glad to have finished my little paper on entropy of geometric structures today, just before the deadline (which is the last day of 2010).
The paper can be seen here in PDF format: Entropy of Geometric Structures.
Abstract: We give a notion of entropy for general gemetric structures, which generalizes well-known notions of topological entropy and geometric entropy of foliations, and which can also be applied to “singular” objects, e.g. singular foliations and Poisson structures. We show some basic properties for this entropy, including the additivity property, analogous to the additivity of Clausius–Boltzmann entropy in physics. In the case of Poisson structures, entropy is a new invariant of dynamical nature, which is related to the transverse structure of the characteristic foliation by symplectic leaves.
This is a short paper (1st version has only 12 pages), but I’m sure it contains lots of interesting ideas and questions for future investigations :-)
This paper is submitted to a special volume of the Bulletin of the Brazilian Mathematical Society dedicated to Poisson geometry.