We show that the measure preserving action of ℤ2 dual to the action defined by the commuting automorphisms ×x and ×y on the discrete group ℤ[x
 ±1,y
 ±1]/〈1+x+y〉ℤ[x
 ±1,y
 ±1] is measurably isomorphic to a ℤ2 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 ℤ2 actions which are measurably isomorphic without being topologically conjugate.

/pdf/almost-block-independence-for-the-three-dot-2-dynamical-4z6y1d6sph.pdf

Almost block independence for the three dot ℤ2 dynamical system

We show that an expansive ℤ2 action on a compact abelian group is measurably isomorphic to a two-dimensional Bernoulli shift if and only if it has completely positive entropy. The proof uses the algebraic structure of such actions described by Kitchens and Schmidt and an algebraic characterisation of theK property due to Lind, Schmidt and the author. As a corollary, we note that an expansive ℤ2 action on a compact abelian group is measurably isomorphic to a Bernoulli shift relative to the Pinsker algebra. A further corollary applies an argument of Lind to show that an expansiveK action of ℤ2 on a compact abelian group is exponentially recurrent. Finally an example is given of measurable isomorphism without topological conjugacy for ℤ2 actions.

/pdf/the-bernoulli-property-for-expansive-2-actions-on-compact-1snw0vm4sp.pdf

The bernoulli property for expansive ℤ2 actions on compact groups

Let G be a compact connected metric abelian group equipped with its normalised Haar measure. Let Tx = a+A(x) be a continuous surjective affine map of G such that ≡ 1 is the only character of G satisfying ◦ A n = for some positive integer n. Then ifis a polynomial with real coefficients mapping the natural numbers to themselves, (pl) 1=1 is the sequence of rational primes and f is in L p (G) for p > 1, we prove that lim N!1 1 N N X n=1

On affine maps and an arithmetic limit on compact groups

We prove that ergodic automorphisms of compact groups are Bernoulli shifts, and that skew products with such automorphisms are isomorphic to direct products. We give a simple geometric demonstration of Yuzvinskii’s basic result in the calculation of entropy for group automorphisms, and show that the set of possible values for entropy is one of two alternatives, depending on the answer to an open problem in algebraic number theory. We also classify those algebraic factors of a group automorphism that are complemented.

The structure of skew products with ergodic group automorphisms

Some results on affine transformations of compact groups

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Some results on affine transformations of compact groups