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Non-commutative skew-product extension dynamical systems
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems , First View
- Published online by Cambridge University Press:
- 10 March 2025, pp. 1-23
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- Article
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Starting from a uniquely ergodic action of a locally compact group G on a compact space
$X_0$, we consider non-commutative skew-product extensions of the dynamics, on the crossed product 
$C(X_0)\rtimes _\alpha {\mathbb Z}$, through a 
$1$-cocycle of G in 
${\mathbb T}$, with 
$\alpha $ commuting with the given dynamics. We first prove that any two such skew-product extensions are conjugate if and only if the corresponding cocycles are cohomologous. We then study unique ergodicity and unique ergodicity with respect to the fixed-point subalgebra by characterizing both in terms of the cocycle assigning the dynamics. The set of all invariant states is also determined: it is affinely homeomorphic with 
${\mathcal P}({\mathbb T})$, the Borel probability measures on the one-dimensional torus 
${\mathbb T}$, as long as the system is not uniquely ergodic. Finally, we show that unique ergodicity with respect to the fixed-point subalgebra of a skew-product extension amounts to the uniqueness of an invariant conditional expectation onto the fixed-point subalgebra.
