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Two-point cocycles with a strong ergodicity property. (English) Zbl 0759.28011
Let $$T: (X,{\mathcal B},\mu)\to(X,{\mathcal B},\mu)$$ be ergodic with discrete spectrum, and $$\phi: X\to G$$ a cocycle. $$T_ \phi: X\times G\to X\times G$$ defined by $$T_ \phi(x,g)=(Tx,\phi(x)+g)$$ is a $$G$$-extension of $$T$$. $$\phi$$ is said to be ergodic if $$T_ \phi$$ is ergodic with respect to $$\mu\times\lambda_ G$$, where $$\lambda_ G$$ is the Haar measure on $$G$$, a compact Abelian group. Let $$S\in C(T)$$, the centralizer of $$T$$ and $$\phi: X\to\mathbb{Z}_ 2$$, then $$\phi$$ is called $$S$$-strongy ergodic if $$\phi S^{j(1)}+\phi S^{j(2)}+\cdots+\phi S^{j(k)}+\phi U$$ is ergodic for every sequence of integers $$j(1)<j(2)<\cdots< j(k)$$, $$k\geq 2$$, and every $$U\in C(T)$$. In this case all $$\mathbb{Z}_ 2\times\mathbb{Z}_ 2\times\cdots$$ extensions $$T_ \psi$$, where $$\psi=\phi\times\phi S\times \phi S^ 2\times\cdots$$, are ergodic. Also, such examples give rise to loosely Bernoulli examples of weakly isomorphic, but non-isomorphic transformations. The second author [Probab. Theory Relat. Fields 78, No. 4, 491-507 (1988; Zbl 0628.60015)] showed that the notion of $$S$$-strong ergodicity is not vacuous, and he asked whether for each ergodic $$\phi: X\to\mathbb{Z}_ 2$$ there exists $$S\in C(T)$$ such that $$\phi$$ is $$S$$-strongly ergodic. That question is answered affirmatively in this note.

##### MSC:
 28D05 Measure-preserving transformations