Speedups of ergodic group extensions (Q2842222)

The authors study relative speedups of \(G\)-extensions of ergodic automorphisms. A speedup of an ergodic automorphism \(T\) on a space \((X,\mathcal{B},\mu)\) has the form \(T^p:x\mapsto T^{p(x)}(x)\) with an integer-valued function \(p\). If \(T\) is a factor of an automorphism \(S\) of a space \((Y,\mathcal{C},\nu)\), then a speedup of \(S\) relative to \(T\) is a speedup \(S^p\), where \(p\) is measurable with respect to the factor \((X,\mathcal{B},\mu)\). The paper deals with the special case where \(S\) is a \(G\)-extension of \(T\), with \(G\) a locally compact second-countable group endowed with the left Haar measure. A \(G\)-extension of \(T\) has the form NEWLINE\[NEWLINE T_{\sigma}(x,g) = (T(x),\sigma(x,1)g),\qquad T_{\sigma}:X \times G \circlearrowleft, NEWLINE\]NEWLINE where \(\sigma:X \times \mathbb{Z} \rightarrow G\) is a measurable cocycle over \(T\). Two \(G\)-extensions \(T_{\sigma}\) and \(T'_{\sigma'}\) on spaces \(X\) and \(X'\) are called \(G\)-isomorphic if there is an isomorphism \(\Phi:X\times G\rightarrow X'\times G\) between \(T_{\sigma}\) and \(T'_{\sigma'}\) of the form NEWLINE\[NEWLINE \Phi(x,g) = (\phi(x),\alpha(x)g), NEWLINE\]NEWLINE where \(\phi:X\rightarrow X'\) is an isomorphism between \(T\) and \(T'\), and \(\alpha:X\rightarrow G\) is a measurable function, called a transfer function. The main result of the paper asserts that, given two \(G\)-extensions \(T_{\sigma}\) and \(T'_{\sigma'}\) of \(T\) and \(T'\), where \(T_{\sigma}\) is ergodic and \(T'\) is aperiodic, there exists a relative speedup of \(T_{\sigma}\) (i.e., relative to the base factor \(T\)), which is \(G\)-isomorphic to \(T'_{\sigma'}\) by a \(G\)-isomorphism whose transfer function \(\alpha\) has values in a given neighborhood of the identity \(e_G\). In the case of a discrete group, the result can be strengthened by demanding that \(\alpha(x)=e_G\) almost everywhere. This theorem is then applied to give necessary and sufficient conditions for two ergodic \(n\)-point or countable extensions to be related in the above way. The authors' main result can be regarded as a conditional version of a result from \textit{P. Arnoux} et al. [Isr. J. Math. 50, 160--168 (1985; Zbl 0558.58019)].