Ergodic properties of real cocycles and pseudo-homogeneous Banach spaces

Lemańczyk, M.; Parreau, F.; Volný, D.

doi:10.1090/S0002-9947-96-01799-0

Ergodic properties of real cocycles and pseudo-homogeneous Banach spaces
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by M. Lemańczyk, F. Parreau and D. Volný

Trans. Amer. Math. Soc. 348 (1996), 4919-4938

DOI: https://doi.org/10.1090/S0002-9947-96-01799-0

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Abstract:

Given an irrational rotation, in the space of real bounded variation functions it is proved that there are ergodic cocycles whose small perturbations remain ergodic; in fact, the set of ergodic cocycles has nonempty dense interior. Given a pseudo-homogeneous Banach space and an irrational rotation, we study the set of elements satisfying the mean ergodic theorem. Once such a space is not homogeneous, we prove it is not reflexive and not separable. In “natural" cases, up to $L^1$-cohomology, the only elements satisfying the mean ergodic theorem are those from the closure of trigonometric polynomials. For pseudo-homogeneous spaces admitting a Koksma’s inequality ergodicity of the corresponding cylinder flows can be deduced from spectral properties of some circle extensions. In particular this is the case of Lebesgue spectrum (in the orthocomplement of the space of eigenfunctions) for the circle extension.

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