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  • Non-existence of a universal zero-entropy system via generic actions of almost complete growth

  • Part of
  • GEORGII VEPREV
  • Journal:
    Ergodic Theory and Dynamical Systems / Volume 44 / Issue 12 / December 2024
    Published online by Cambridge University Press:
    12 April 2024, pp. 3630-3644
    Print publication:
    December 2024
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  • We prove that a generic probability measure-preserving (p.m.p.) action of a countable amenable group G has scaling entropy that cannot be dominated by a given rate of growth. As a corollary, we obtain that there does not exist a topological action of G for which the set of ergodic invariant measures coincides with the set of all ergodic p.m.p. G-systems of entropy zero. We also prove that a generic action of a residually finite amenable group has scaling entropy that cannot be bounded from below by a given sequence. In addition, we show an example of an amenable group that has such a lower bound for every free p.m.p. action.