This chapter discusses the entropy functional for dynamical systems and their random perturbations. The chapter formulates the Gibbs variational principle that unifies the Donsker-Varadhan theory for Markov chains, the equilibrium classical statistical mechanics of lattice systems and the theories for symbolic dynamics, expanding maps and Anosov diffeomorphisms. The chapter discusses the relation between the quantities for dynamical systems and the non-random limits of the corresponding quantities given for their random perturbations. The case of symbolic dynamics is also discussed. It is shown in the typical cases that the two functionals f and q coincide with each another and the result of the computation of them may be interpreted as a characterization of the metrical entropy. There is a discussion about the case of general maps of intervals. The counterexample leads to the conjecture that the entropy functional f is affine when the underlying dynamical system (X, F) is structurally stable. In a special case this conjecture is proved to be affirmative and the structurally stable systems are Morse-Smale systems. © 1984, Kinokuniya Company Ltd.
Takahashi, Y. (1984). Entropy Functional (free energy) for Dynamical Systems and their Random Perturbations. North-Holland Mathematical Library, 32(C), 437–467. https://doi.org/10.1016/S0924-6509(08)70404-5