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Ergodic invariant measures for group-extensions of dynamical systems. (Mesures invariantes ergodiques pour des produits gauches.) (French) Zbl 1155.28009
Let \((X,{\mathcal X})\) be a measurable space and let \(\tau\) be a bi-measurable bijection from \(X\) onto \(X\). Let \(\varphi\) be a measurable function from \(X\) into a second countable locally compact group \(G\). Define the transformation \(\tau_\varphi\) on \(X \times G\) by \(\tau_\varphi(x,g) = (\tau(x), g \, \varphi(x))\). Let \(\lambda\) be a \(\sigma\)-finite measure on \(X \times G\) which can be written as \(\lambda(dx, dg) = \mu(dx) \, N(x,dg)\), where \(\mu\) is a probability measure on \((X,{\mathcal X})\) and \(N\) is a positive Radon kernel from \((X,{\mathcal X})\) into \((G,{\mathcal B}(G))\). The author gives a description of the measure \(\lambda\) if \(\lambda\) is \(\tau_\varphi\)-invariant and ergodic. In addition he obtains a generalization of the cocycle reduction theorem of O. Sarig [Invent. Math. 157, No. 3, 519–551 (2004; Zbl 1052.37004)] taking values in a second countable locally compact group.

MSC:
28D05 Measure-preserving transformations
37A05 Dynamical aspects of measure-preserving transformations
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
37A40 Nonsingular (and infinite-measure preserving) transformations
Citations:
Zbl 1052.37004
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