Recurrence, dimension and entropy

Let (Sigma (A), T) be a topologically mixing subshift of finite type on an alphabet consisting of m symbols and let Phi:Sigma (A)-->R-d be a continuou s function. Denote by sigma (Phi)(x) the ergodic limit lim(n --> infinity)n (-1)Sigma (n-1)(j=0)Phi (T(j)x) when the limit exists. Possible ergodic lim its are just mean values integral Phid mu for all T-invariant measures. For any possible ergodic limit alpha, the following variational formula is pro ved: h(top)({x is an element of Sigma (A):sigma (Phi)(x) = alpha}) = sup{h(mu):i ntegral Phid mu = alpha} where h(mu) denotes the entropy of mu and h(top) denotes topological entrop y. It is also proved that unless all points I ave the same ergodic limit, t hen the set of points whose ergodic limit does not exist has the same topol ogical entropy as the whole space Sigma (A).