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On special representations of automorphisms of \(\sigma \)-finite measure spaces using Poincare recurrence theorem and Hopf decomposition

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Abstract

We extend and strengthen Hopf’s decomposition theorem for \(\sigma \)-finite measure spaces, and provide a purely constructive proof. From this investigation, the Poincaré recurrence theorem follows in the more general setting. We prove that this theorem holds for (the restriction of a given automorphism onto) the conservative part and may fail for the dissipative part of the respective Hopf decomposition of the measure space. Our method of proof allows to view the restriction of a given automorphism on the dissipative part as the so-called direct shift automorphism, whereas the restriction onto the conservative part admits various special representations linked with an arbitrary subset of positive measure.

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Acknowledgements

Using this opportunity, the authors express their gratitude to Alexei Ber (Tashkent, Uzbekistan), Dmitriy Zanin (UNSW, Sydney) and Thomas Scheckter (UNSW, Sydney) for constructive comments.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Kensington, NSW, 2052, Australia

    Fedor Sukochev

  2. Institute of Mathematics named V. I. Romanovsky, Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan

    Aleksandr Veksler

Authors
  1. Fedor Sukochev
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  2. Aleksandr Veksler
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Correspondence to Fedor Sukochev.

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Communicated by Timur Oikhberg.

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Sukochev, F., Veksler, A. On special representations of automorphisms of \(\sigma \)-finite measure spaces using Poincare recurrence theorem and Hopf decomposition. Adv. Oper. Theory 8, 46 (2023). https://doi.org/10.1007/s43036-023-00279-5

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  • DOI: https://doi.org/10.1007/s43036-023-00279-5

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