Entropy and large deviation

Abstract

The author shows the existence of a deviation function for the maximal measure mu of a hyperbolic rational map of degree d. He relates several results of large deviation with the thermodynamic formalism of ergodic theory. The maximal measure plays a distinguished role among other invariant measures, because the stochastic process given by the rational map and the maximal measure will generate a free energy function, whose Legendre transform in the set of invariant measures will be log d minus the entropy in the sense of Shannon-Kolmogorov. This result is associated with the relation between pressure and free energy. A general description of the result is as follows. Consider mu to be the maximal entropy measure and v another invariant measure. The ergodic theorem claims that the mean of the sum of Dirac measures in the orbit of a mu -almost everywhere point z, will converge to mu . Given a convex neighbourhood G of v in the set of measures, the author estimates the deviations of the mean of the sum of Dirac measures in the orbit of a mu -almost everywhere point z, with respect to this neighbourhood G. If the neighbourhood G is very small and he considers large iterates, the exponential value of decreasing of the mu -measure of points whose mean orbit is in G approximately the entropy of v minus log d. In this way, he calculates the entropy of v as an information of large deviation related to the maximal measure mu . He applies this result, using a contraction principle, to measure the deviation of the Liapunov number of the maximal measure. The same proof presented in this paper also works (with minor modifications) for shifts of finite type in the lattice N.