Gabor (Trans. Amer. Math. Soc. 377, 2024; arXiv:1909.11453, 2019) proved that a finitary factor of an i.i.d. process need not be finitarily isomorphic to an i.i.d. process, refuting a conjecture of Smorodinsky, and constructed such a factor with no finitary i.i.d. factors of entropy greater than log 2. The paper's first open problem asks whether this entropy threshold can be removed entirely: does there exist a finitary factor of an i.i.d. process none of whose finitary factors of positive entropy is an i.i.d. process? The same paper's Proposition 2.2 shows that its own obstruction criterion (an exponential block-frequency decay bound) is satisfied by every finitary factor of i.i.d. at some positive entropy level, so a positive answer requires a genuinely new obstruction, while a negative answer would be a small-entropy finitary analogue of the classical theorem that factors of Bernoulli shifts are Bernoulli. Two descriptive-set-theoretic questions of Tim Austin from the same section (Borelness of the finitary-isomorphism and finitary-factoring sets in the weak* space of shift-invariant measures) also remain open and are recorded as adjacent problems.

Background

A stationary process is a shift-invariant probability measure μ\mu on the sequence space AZA^{\mathbb{Z}} over a finite alphabet AA; its Kolmogorov–Sinai entropy h(μ)h(\mu) measures the information produced per time step. An i.i.d. process (Bernoulli shift) is the product measure p×Zp^{\times\mathbb{Z}} on BZB^{\mathbb{Z}}. A factor map from a process XX to a process YY is a measurable, shift-equivariant map carrying the distribution of XX to that of YY; it is called finitary if, for almost every realization xx of XX, the zero coordinate of the image is determined by some finite window of xx around the origin (an almost surely finite coding radius, equivalently almost sure continuity of the map). Two processes are finitarily isomorphic when there is an isomorphism that is finitary in both directions. Classical Ornstein theory establishes, in the purely measurable category, that two Bernoulli shifts of equal entropy are isomorphic, that factors of Bernoulli shifts are Bernoulli, and (Sinai's weak isomorphism theorem) that every ergodic system (one whose shift-invariant events have probability 0 or 1) of finite entropy factors measurably onto every Bernoulli system of no greater entropy. Keane and Smorodinsky proved the finitary counterpart of the isomorphism theorem — any two i.i.d. processes of equal entropy are finitarily isomorphic — which suggested a program of finding finitary counterparts for the rest of Ornstein theory. Rudolph asked whether every finitary factor of an i.i.d. process is finitarily isomorphic to an i.i.d. process, and Smorodinsky conjectured a positive answer, having settled the case where the factor map is continuous (a fixed finite window) and the case of finitely dependent processes (processes whose coordinate sets separated by more than some fixed gap are independent). Gabor refuted this conjecture. His Theorem 1.1 gives a finitary factor of an i.i.d. process that is not finitarily isomorphic to any i.i.d. process, and his quantitative Theorem 1.2 strengthens it: there is a process of entropy greater than log⁡2\log 2 that is a finitary factor of an i.i.d. process but has no finitary i.i.d. factors of entropy greater than log⁡2\log 2. His Theorem 1.3 shows that any ergodic system of positive entropy is isomorphic to a process none of whose finitary factors is an i.i.d. process — so processes with no finitary i.i.d. factors at all exist in abundance — but that construction does not yield a finitary factor of an i.i.d. process; as a corollary it also answers negatively the question, attributed by Gabor to G. Pete and T. Austin, of whether a general finitary weak Pinsker property holds (the weak Pinsker property, proved by Austin in the measurable category, splits every ergodic system as a product of a Bernoulli system and a low-entropy system). The paper's obstruction tool is Lemma 2.1: if an ergodic process finitarily factors onto an i.i.d. process of entropy h′h', then a certain exponential decay bound on the frequencies of long blocks must hold in the source. The crucial accompanying result, Proposition 2.2, shows that every finitary factor of an i.i.d. process satisfies this bound for some h′>0h' > 0, so this criterion cannot witness the absence of all finitary i.i.d. factors; the author notes explicitly that it may even be true that every finitary factor of an i.i.d. process does finitarily factor onto some i.i.d. process. The paper's Open Problems section also poses two descriptive-set-theoretic questions suggested by Tim Austin — whether, in the Polish space (a complete separable metric space) of shift-invariant measures with the weak* topology, the set of measures finitarily isomorphic to a fixed i.i.d. process, and the set of measures finitarily factoring onto it, are Borel sets; these are adjacent open problems recorded alongside, but distinct from, the core question below.

Problem Statement

Does there exist a positive-entropy finitary factor of an i.i.d. process none of whose finitary factors of positive entropy is an i.i.d. process? (Open Problem (i) of Gabor's 'On the failure of Ornstein theory in the finitary category'; the source adds 'regardless of the factor's entropy' — every process trivially admits the one-point, zero-entropy i.i.d. factor, and any finitary factor of an i.i.d. process is ergodic, for which every zero-entropy i.i.d. factor is trivial, so the meaningful reading is positive entropy throughout the task: the witness factor is required to have positive entropy, and the excluded i.i.d. factors are those of positive entropy.) Equivalently: can the entropy threshold log⁡2\log 2 in Gabor's Theorem 1.2 be pushed to zero — is there a stationary process XX of positive entropy that is a finitary factor of some i.i.d. process and has no positive-entropy finitary i.i.d. factor — or does every finitary factor of an i.i.d. process of positive entropy admit a finitary factor map onto some i.i.d. process of positive entropy? Note that by Proposition 2.2 of the source, a positive answer cannot be certified through the Lemma 2.1 exponential block-decay obstruction, which every finitary factor of i.i.d. satisfies at some positive entropy level; an answer that only lowers the threshold from log⁡2\log 2 to some other positive constant, or that produces a non-finitary factor of i.i.d. (as Theorem 1.3 already does), does not settle the question.

References

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  2. Tom Meyerovitch, Yinon Spinka, Entropy-efficient finitary codings, Journal of Modern Dynamics 20 (2024), 1–49, DOI 10.3934/jmd.2024001, https://doi.org/10.3934/jmd.2024001 (preprint arXiv:2201.06542, https://arxiv.org/abs/2201.06542)
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