Abstract
Exploiting a spectral criterion for a system not to be AT we give some new examples of zero entropy systems without the AT property. Our examples include those with finite spectral multiplicity—in particular we show that the system arising from the Rudin–Shapiro substitution is not AT. We also show that some nil-rotations on a quotient of the Heisenberg group as well as some (generalized) Gaussian systems are not AT. All known examples of non AT-automorphisms contain a Lebesgue component in the spectrum.
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Communicated by K. Schmidt.
Research partially supported by Polish MNiSzW grant N N201 384834, Marie Curie “Transfer of Knowledge” EU program—project MTKD-CT-2005-030042 (TODEQ) and MSRI (Berkeley) program “Ergodic Theory and Additive Combinatorics”.
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El Abdalaoui, E.H., Lemańczyk, M. Approximate transitivity property and Lebesgue spectrum. Monatsh Math 161, 121–144 (2010). https://doi.org/10.1007/s00605-010-0223-y
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DOI: https://doi.org/10.1007/s00605-010-0223-y