Non-commutative skew-product extension dynamical systems

2024/10/08 by Vitonofrio Crismale, Crismale, Vitonofrio, Simone Del Vecchio +5
Computer Science · Mathematics · #Cellular Automata and Applications #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2410.07255

openalex publication_date 2024/10/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Starting from a uniquely ergodic action of a locally compact group G on a compact space X₀, we consider non-commutative skew-product extensions of the dynamics, on the crossed product C(X₀)\rtimes_αℤ, through a 1-cocycle of G in \mathbbT, 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 P(\mathbbT), the Borel probability measures on the one-dimensional torus \mathbbT, 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

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