Non-commutative skew-product extension dynamical systems
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912533683634176 |
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| author | Crismale, Vitonofrio Del Vecchio, Simone Griseta, Maria Elena Rossi, Stefano |
| author_facet | Crismale, Vitonofrio Del Vecchio, Simone Griseta, Maria Elena Rossi, Stefano |
| contents | 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 $C(X_0)\rtimes_α\mathbb{Z}$, through a $1$-cocycle of $G$ in $\mathbb{T}$, with $α$ commuting with the given dynamics. We first prove that any such two skew-product extensions are conjugate if and only if the corresponding cocycles are cohomologous. We then study unique ergodicity and unique ergodicity w.r.t. 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 w.r.t. the fixed-point subalgebra of a skew-product extension amounts to the uniqueness of an invariant conditional expectation onto the fixed-point subalgebra |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07255 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-commutative skew-product extension dynamical systems Crismale, Vitonofrio Del Vecchio, Simone Griseta, Maria Elena Rossi, Stefano Dynamical Systems Operator Algebras 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 $C(X_0)\rtimes_α\mathbb{Z}$, through a $1$-cocycle of $G$ in $\mathbb{T}$, with $α$ commuting with the given dynamics. We first prove that any such two skew-product extensions are conjugate if and only if the corresponding cocycles are cohomologous. We then study unique ergodicity and unique ergodicity w.r.t. 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 w.r.t. the fixed-point subalgebra of a skew-product extension amounts to the uniqueness of an invariant conditional expectation onto the fixed-point subalgebra |
| title | Non-commutative skew-product extension dynamical systems |
| topic | Dynamical Systems Operator Algebras |
| url | https://arxiv.org/abs/2410.07255 |