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Invariant Conditional Expectations and Unique Ergodicity for Anzai Skew-Products

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Abstract

Anzai skew-products are shown to be uniquely ergodic with respect to the fixed-point subalgebra if and only if there is a unique conditional expectation onto such a subalgebra which is invariant under the dynamics. For the particular case of skew-products, this solves a question raised by B. Abadie and K. Dykema in the wider context of \(C^*\)-dynamical systems.

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Notes

  1. The different role played by “continuous” and “measurable non-continuous functions” in the investigation of some relevant ergodic properties was firstly remarked in [9], see also [13]. The reader is further referred to [6, 7] for the generalisation to the noncommutative situation, and in particular to [3] for the noncommutative torus.

  2. In [8], Theorem 2.1, it was also shown that the a-priori weaker condition (v), also characterises the unique ergodicity w.r.t. the fixed-point subspace.

  3. The case \(n=1\) corresponds to the trivial homeomorphism \(\hbox {id}_{(X_o\times \mathbb {T})}\) leading to the trivial fixed-point subalgebra \(C(X_o\times \mathbb {T})\) and trivial conditional expectation \({\mathcal {E}}_1=\hbox {id}_{C(X_o\times \mathbb {T})}\).

  4. To be more precise, whenever g is a solution of (2.3) at the level n, \(h(x, z):= g(x)z^n\) is \(\mu _o\times m\)-equivalent to a \(\Phi _{\theta _o, f}\)-invariant continuous function if and only if \(g=\pi _{\mu _o}(G)\), and G satisfies the twin equation (2.4).

  5. The case \(k=1\) corresponds to the trivial case when \(M_k(C(\mathbb {T}))=C(\mathbb {T})\) and \(\pi _1=\hbox {id}_{C(\mathbb {T})}\).

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Acknowledgements

S.D.V. and F.F. acknowledge the “MIUR Excellence Department Project” awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

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  1. Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via Edoardo Orabona 4, Bari, 70125, Italy

    Simone Del Vecchio & Stefano Rossi

  2. Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, Roma, 00133, Italy

    Francesco Fidaleo

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  1. Simone Del Vecchio
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  2. Francesco Fidaleo
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Correspondence to Simone Del Vecchio.

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Del Vecchio, S., Fidaleo, F. & Rossi, S. Invariant Conditional Expectations and Unique Ergodicity for Anzai Skew-Products. Results Math 78, 15 (2023). https://doi.org/10.1007/s00025-022-01785-3

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