RECURRENCE, DIMENSION AND ENTROPY

Abstract

Let ([sum ]A, T) be a topologically mixing subshift of finite type on an alphabet consisting of m symbols and let Φ:[sum ]A → Rd be a continuous function. Denote by σΦ(x) the ergodic limit limn→∞n−1 [sum ]n−1j=0 Φ(Tjx) when the limit exists. Possible ergodic limits are just mean values ∫ Φdμ for all T-invariant measures. For any possible ergodic limit α, the following variational formula is proved:[formula here]where hμ denotes the entropy of μ and htop denotes topological entropy. It is also proved that unless all points have the same ergodic limit, then the set of points whose ergodic limit does not exist has the same topological entropy as the whole space [sum ]A