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An answer to a question by J. Milnor. (English)
Comment. Math. Helv. 60, 173-178 (1985).
Fix a finite alphabet X and endow the space $M=Z\sp{{\bbfZ}}$ with the topology of direct product. Let S be the shift in M. Given a function $f: X\sp{2r+1}\to X$ define T by $(x\sb n:\quad n\in {\bbfZ})\mapsto (f(x\sb{n-r},...,x\sb{n+r}):\quad n\in {\bbfZ}).$ Clearly S and T commute. Fix a probabilistic measure $μ$ on M ergodic and invariant under all transformations $S\sp mT\sp n$. Denote by $h\sb{m,n}$, m,n$\in {\bbfZ}$, the entropy of $S\sp mT\sp n$ with respect to $μ$ and assume that all $h\sb{m,n}$ are finite. The author gives an answer to the following question posed by J. Milnor. Fix an irrational positive number $ω\sb 0$ and sequences $(m\sb i:$ $i\in {\bbfN})$ and $(n\sb i:$ $i\in {\bbfN})$ of integers such that $m\sb i\to \infty$, $n\sb i\to \infty$, and $m\sb i/n\sb i\to ω\sb 0$. Is it true that the limit $\lim\sb{i\to \infty}(m\sp 2\sb i+n\sp 2\sb i)\sp{-1/2}h\sb{m\sb i,n\sb i}$ exists? The answer is "yes". Moreover, it turns out that this limit does not depend on the sequences $(m\sb i:$ $i\in {\bbfN})$ and $(n\sb i:$ $i\in {\bbfN})$. 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.
Reviewer: W.Jarczyk