Gibbs entropy and dynamics

Let M be the phase space of a physical system. The dynamics is determined by the map T : M-->M, preserving the measure mu. Let nu be another measure on M, dnu=rho dmu. Gibbs introduced the quantity s(rho)=-integralrho log rho dmu as an analog of the thermodynamical entropy. Attempts to reach a closer analogy between thermodynamical and Gibbs entropy lead to the idea to modify the last one and to replace it by the so-called coarse-grained entropy. The dynamics transforms nu in the following way: numapsto]nu(n), dnu(n)=rho composite functionT(-n)dmu. Hence, we obtain the sequence of densities rho(n)=rho composite functionT(-n) and the corresponding values of the Gibbs and the coarse-grained entropy. We discuss the following question: To what extent the Gibbs and coarse-grained entropy are physical? More precisely: (1) do they grow under the dynamics, generated by T? (2) What properties of the dynamics are responsible for this growth? (3) To what extent can this growth be independent of arbitrariness in the construction of the coarse-grained entropy?