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Odometer actions of the Heisenberg group

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Abstract

Let H 3(ℝ) be the 3-dimensional real Heisenberg group. Given a family of lattices Γ1 ⊃ Γ2 ⊃ … ⊂ H 3(ℝ), let T be the associated uniquely ergodic H 3(ℝ)-odometer, i.e., the inverse limit of the H 3(ℝ)-actions by rotations on the homogeneous spaces H 3(ℝ)/Γ j , j ∈ ℕ. We describe explicitly the decomposition of the underlying Koopman unitary representation of H 3(ℝ) into a countable direct sum of irreducible components and find the ergodic 2-fold self-joinings of T. We show that in general, the H 3(ℝ)-odometers are neither isospectral nor spectrally determined.

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Authors and Affiliations

  1. Institute for Low Temperature Physics & Engineering, National Academy of Sciences of Ukraine, 47 Lenin Ave., Kharkov, 61164, Ukraine

    Alexandre I. Danilenko

  2. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100, Toruń, Poland

    Mariusz Lemańczyk

  3. Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-950, Warsaw, Poland

    Mariusz Lemańczyk

Authors
  1. Alexandre I. Danilenko
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  2. Mariusz Lemańczyk
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Corresponding author

Correspondence to Alexandre I. Danilenko.

Additional information

The second named author was supported in part by Narodowe Centrum Nauki grant DEC-2011/03/B/ST1/00407.

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Danilenko, A.I., Lemańczyk, M. Odometer actions of the Heisenberg group. JAMA 128, 107–157 (2016). https://doi.org/10.1007/s11854-016-0003-2

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  • DOI: https://doi.org/10.1007/s11854-016-0003-2

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