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We show that any finite-entropy, countable-valued finitary factor of an i.i.d. process can also be expressed as a finitary factor of a finitevalued i.i.d. process whose entropy is arbitrarily close to the target process. As an application, we give an affirmative answer to a question of van den Berg and Steif [27] about the critical Ising model on Zd . En route, we prove several results about finitary isomorphisms and finitary factors. Our results are developed in a new framework for processes invariant to a permutation group of a countable set satisfying specific properties. This new framework includes all “classical” processes over countable amenable groups and all invariant processes on transitive amenable graphs with “uniquely centered balls”. Some of our results are new already for Z-processes. We prove a relative version of Smorodinsky’s isomorphism theorem for finitely dependent Z-processes. We also extend the Keane–Smorodinsky finitary isomorphism theorem to countable-valued i.i.d. processes and to i.i.d. processes taking values in a Polish space.

Original languageAmerican English
Pages (from-to)1-49
Number of pages49
JournalJournal of Modern Dynamics
StatePublished - 1 Jan 2024


  • amenable
  • entropy
  • Finitary factor
  • finitary isomorphism
  • finitely dependent
  • unimodular

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics


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