Some Large Deviation Results for Dynamical Systems

We prove some large deviation estimates for continuous maps of compact metric spaces and apply them to attractors in differentiable dynamics, rate of escape problems, and to shift spaces. Introduction Consider a discrete time dynamical system generated by a self-map /: X O of some domain X. Let m be a reference measure on X, and let 0 denote the accepted margin of error, and let « = xêLYlS Then mBn -»0 as « —► co. We wish to know if mBn « em for some a, or at least if we can find a and ß so that eß" < mBn < e"" . We are particularly interested in exponential convergence, i.e., a < 0. A different situation, but one that involves the same set of ideas, is the following: Let / be a continuous map or flow, and let A c X be an invariant set that is not an attractor. Because of the invariance of A, if a point is near A then its next few iterates are not likely to be far away. We are interested in the rate of escape from a neighborhood of A. In the case of a flow, this rate also measures the capacity of A as a barrier to transport. More precisely, we let U be a neighborhood of A, define Cn = {xeU:x,fx,...,fxGU} and ask if mCn « enn . Large deviation questions have been successfully dealt with for various stochastic processes (see e.g., [E, S, V]). In the case of dynamical systems, one does not expect nice, explicit rate functions in general, especially when the trajectories do not have good statistical properties. One can, however, ask how Received by the editors June 7, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 58F11.