Entropy-efficient finitary codings

Tom Meyerovitch; Yinon Spinka

doi:10.3934/jmd.2024001

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2024, Volume 20: 1-49. Doi: 10.3934/jmd.2024001

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Entropy-efficient finitary codings

Tom Meyerovitch^1, and
Yinon Spinka^2,

1.
Department of Mathematics, Ben Gurion University of the Negev, Be'er Sheva, 8410501, Israel
2.
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada and School of Mathematical Sciences, Tel Aviv University, Tel Aviv 6997801, Israel

Received: October 05, 2022

Revised: July 14, 2023

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Abstract

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 finite-valued 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 $ \mathbb{Z}^d $. 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 $ \mathbb{Z} $-processes. We prove a relative version of Smorodinsky's isomorphism theorem for finitely dependent $ \mathbb{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.

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Mathematics Subject Classification: Primary: 37A35; Secondary: 60G20, 60G60.

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Figure 1. An illustration of the relations in an $ (M, X', \tilde X') $-adapted partial isomorphism. The arrows represent finitary M-block factors

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