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On regular generators of \({\mathbb{Z}}^ 2\)-actions in exhaustive partitions. (English) Zbl 0622.28015

Let G be a free Abelian group of rank 2 of automorphisms of a Lebesgue probability space \((X,\beta,\mu).\) The quadruple \((X,\beta,\mu,G)\) (denoted \((X,G))\) is said to be a two-dimensional dynamical system (or \({\mathbb{Z}}^ 2\)-action). The entropy theory for such systems was developed by J. P. Conze [Z. Wahrscheinlichkeitstheor. Verw. Gebiete 25, 11- 30 (1972; Zbl 0261.28015)], Y. Katznelson and B. Weiss [Isr. J. Math. 12, 161-173 (1972; Zbl 0239.28014)] and the theory of invariant partitions was developed by the author [Bull. Acad. Pol. Sci., Ser. Sci. Math. 29, 349-362 (1981; Zbl 0479.28016)].
Denote by b(G) the set of all ordered pairs of independent generators of G and write \(\pi =\{(i,j)\in {\mathbb{Z}}^ 2:(i,j)\prec (0,0)\}\) (where \(\prec\) is the lexicographical ordering of \({\mathbb{Z}}^ 2)\). Let \((T,S)\in b(G)\) and P be a measurable partition with finite entropy and define \[ P_{\bar G}=\bigvee_{(k,\ell)\in \pi}T^ kS^{\ell}P,\quad P_ G=\bigvee_{(k,\ell)\in {\mathbb{Z}}^ 2}T^ kS^{\ell}P, \] then P is a generator for (X,G) if \(P_ G=\epsilon\) (the measurable partition of X into single points). Denote by \(B_ G\) the set of all generators of \((X,G)\) with finite entropy and for \(P\in B_ G\) write \(\zeta_ P=P\vee P_{\bar G}.\) The purpose of this paper and an earlier paper (with M. Kobus, to appear) is to investigate the question of whether the partition \(\zeta_ P\) is \((T,S)\) perfect for any \(P\in B_ G;\) the analogous result for single automorphisms has a positive answer [V. A. Rokhlin and Ya. G. Sinai [Dokl. Akad. Nauk SSSR 141, 1038-1041 (1961; Zbl 0161.343)] but is negative in this case. Call such a generator regular. The author shows that for every totally ergodic \({\mathbb{Z}}^ 2\)-action with finite entropy there exists a regular generator in a given exhaustive partition and the set of regular generators is dense in the set of all generators. The regular generators seem to be useful in characterizing those groups with zero entropy, in a similar manner to the situation for single automorphisms.
Reviewer: G.R.Goodson

MSC:

28D20 Entropy and other invariants
28D15 General groups of measure-preserving transformations
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