[For the entire collection see Zbl 0511.00026.]
Let \(X\) be a compact metric space with distance function \(d\) and \(f:X\to X\) be a continuous mapping preserving a Borel probability non-atomic measure \(m\). Assume that \(h_ m(f)\), the entropy of the \(f\), is finite. Denote as usual for \(x,y\in X\) and a positive integer \(n\), \(d^ f_ n(x,y) = \max_{0\leq i\leq n-1} d(f^ i x, f^ i y)\) and for \(r>0\) let \(B^ f_ n(x,r)\) be the \(d^ f_ n\)-ball about \(x\) of radius \(r\). Then the proof of the following theorem is given in the paper:
For \(m\)-almost every \(x\in X,\) \[ \begin{aligned} &\lim_{\delta\to 0} \liminf_{n\to\infty} \frac{-\log m(B^ f_ n (x,\delta))}{n} = \lim_{\delta\to 0} \limsup_{n\to\infty} \frac{-\log m(B^ f_ n(x,\delta))}{n} \overset{\text{def}}= h_ m(f,x); \tag{a}\\ &h_ m(f,x) \text{ is \(f\)-invariant;}\tag{b}\\ &\int_{X} h_ m(f,x) dm = h_ m(f).\tag{c} \end{aligned} \]