## 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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