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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.

28D05 Measure-preserving transformations