# zbMATH — the first resource for mathematics

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
##### Keywords:
ergodic invariant measure; cocycle reduction theorem
Zbl 1052.37004
Full Text: