Recurrence, Dimension and Entropy

Let ([sum ] A , T ) be a topologically mixing subshift of finite type on an alphabet consisting of m symbols and let Φ:[sum ] A → R d be a continuous function. Denote by σ Φ ( x ) the ergodic limit lim n →∞ n −1 [sum ] n −1 j =0 Φ( T j x ) when the limit exists. Possible ergodic limits are just mean values ∫ Φ d μ for all T -invariant measures. For any possible ergodic limit α, the following variational formula is proved: [formula here] where h μ denotes the entropy of μ and h top 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