2022/01/17 by Tom Meyerovitch, Meyerovitch, Tom, Yinon Spinka +1
Computer Science · Mathematics · #11H06 #28D99 #37A35 #60G10 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2201.06542
openalex publication_date 2022/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.