Coarse-grained entropy in dynamical systems

Let $M$ be the phase space of a physical system. Consider the dynamics, determined by the invertible map $T: M \to M$, preserving the measure $\mu$ on $M$. Let $\nu$ be another measure on $M$, $d\nu = \rho d\mu$. Gibbs introduced the quantity $s(\rho) = − \int \rho \log \rho d \mu$ as an analog of the thermodynamical entropy. We consider a modification of the Gibbs (fine-grained) entropy the so called coarse-grained entropy.
First we obtain a formula for the difference between the coarse-grained and Gibbs entropy. The main term of the difference is expressed by a functional usually referenced to as the Fisher information.
Then we consider the behavior of the coarse-grained entropy as a function of time. The dynamics transforms $\nu$ in the following way: $\nu \mapsto \nu_n$, $d\nu_n = \rho \circ T^{-n} d\mu$. Hence, we obtain the sequence of densities $\rho_n = \rho \circ T^{-n}$ and the corresponding values of the Gibbs and the coarse-grained entropy. We show that while the Gibbs entropy remains constant, the coarse-grained entropy has a tendency to a growth and this growth is determined by dynamical properties of the map $T$. Finally, we give numerical calculation of the coarse-grained entropy as a function of time for systems with various dynamical properties: integrable, chaotic and with mixed dynamics and compare these calculation with theoretical statements.