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Almost block independence for the three dot \(\mathbb Z^ 2\) dynamical system. (English) Zbl 0790.28013
Summary: We show that the measure preserving action of \(\mathbb Z^ 2\) dual to the action defined by the commuting automorphisms \(\times x\) and \(\times y\) on the discrete group \(\mathbb Z[x^{\pm 1},y^{\pm 1}]/\langle1+ x+ y\rangle\mathbb Z[x^{\pm 1},y^{\pm 1}]\) is measurably isomorphic to a \(\mathbb Z^ 2\) Bernoulli shift. This was conjectured in recent work by Lind, Schmidt and the author, where it was shown that this action has completely positive entropy. An example is given of \(\mathbb Z^ 2\) actions which are measurably isomorphic without being topologically conjugate.

MSC:
37A15 General groups of measure-preserving transformations and dynamical systems
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
37B99 Topological dynamics
28D15 General groups of measure-preserving transformations
28D10 One-parameter continuous families of measure-preserving transformations
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