Group extensions of Gibbs-Markov maps
Let \((X,\mathcal{B},m,T,\alpha)\) be an exact probability-preserving Markov map, where \((X,\mathcal{B},m)\) is some probability space, \(T\colon X \to X\) a probability-preserving transformation and \(\alpha\) denotes a generating Markov partition. For each \(a\in \bigvee_{i=0}^{n-1}T^{-i}\alpha ,\) let \(\nu_a\) be the local inverse of the map \(T^n:a\to T^na,\) and denote by \(\nu_a'\) the Radon-Nikodym derivantive of \(m\circ \nu_a\) with respect to \(m.\) The map \(T\) is said to be a Gibbs-Markov map if \(\inf_{a\in \alpha }m(Ta)\) and if \(T\) has the Gibbs-property [see e.g., the authors, Stoch. Dyn. 1, 193-237 (2001)]. Let \(G\) be a locally compact, abelian, secound countable group, and \(\phi:X\to G\) a measurable cocycle. Consider the corresponding skew-product \(T_{\phi}:X\times G\to X\times G\) defined by \(T_{\phi}(x,y)=(Tx,y+\phi (x))\), and which is invariant under the product measure. In this paper the authors give different characterizations for \(T_{\phi}\) to be exact. For example they show that \[ T_{\phi} \text{ is exact } \iff \phi \text{ is aperiodic } \iff T_{\phi} \text{ is weakly mixing}. \] Other characterizations are also given.
scientific article; zbMATH DE number 1837116