An answer to a question by J. Milnor
Fix a finite alphabet X and endow the space \(M=Z^{{\mathbb{Z}}}\) with the topology of direct product. Let S be the shift in M. Given a function \(f: X^{2r+1}\to X\) define T by \((x_ n:\quad n\in {\mathbb{Z}})\mapsto (f(x_{n-r},...,x_{n+r}):\quad n\in {\mathbb{Z}}).\) Clearly S and T commute. Fix a probabilistic measure \(\mu\) on M ergodic and invariant under all transformations \(S^ mT^ n\). Denote by \(h_{m,n}\), m,n\(\in {\mathbb{Z}}\), the entropy of \(S^ mT^ n\) with respect to \(\mu\) and assume that all \(h_{m,n}\) are finite. The author gives an answer to the following question posed by J. Milnor. Fix an irrational positive number \(\omega_ 0\) and sequences \((m_ i:\) \(i\in {\mathbb{N}})\) and \((n_ i:\) \(i\in {\mathbb{N}})\) of integers such that \(m_ i\to \infty\), \(n_ i\to \infty\), and \(m_ i/n_ i\to \omega_ 0\). Is it true that the limit \(\lim_{i\to \infty}(m^ 2_ i+n^ 2_ i)^{-1/2}h_{m_ i,n_ i}\) exists? The answer is ''yes''. Moreover, it turns out that this limit does not depend on the sequences \((m_ i:\) \(i\in {\mathbb{N}})\) and \((n_ i:\) \(i\in {\mathbb{N}})\). This assertion (Theorem 2) is deduced from a more general result (Theorem 1). Also some remarks concerning a general situation of two commuting automorphisms of a Lebesgue space are made.
commuting automorphisms of a Lebesgue space
scientific article; zbMATH DE number 3955309