Kalikow's uniform martingale questions: finite-alphabet and minimal-alphabet representations of measure-preserving transformations

Kalikow's paper 'Infinite partitions and Rokhlin towers' (2012) proves that every non-periodic measure-preserving transformation extends to a uniform martingale — a stationary process whose conditional law of the present symbol given the past converges uniformly over all pasts — on a countable alphabet, and that the extension can be made an isomorphism when the transformation is a product with a nontrivial Bernoulli shift. The paper poses four open questions about compressing these representations: (1) can every finite-alphabet transformation be extended to a uniform martingale on a finite alphabet, or at least on a countable alphabet whose letter distribution has finite entropy; (2) is every positive-entropy finite-alphabet transformation isomorphic to a finite-alphabet uniform martingale, in particular when it is a product of a Bernoulli shift with another transformation; (3) is every transformation isomorphic to a countable-alphabet uniform martingale even without the weak Pinsker hypothesis; (4) must a uniform martingale on a three-letter alphabet with entropy below $\log 2$ be isomorphic to, or at least extendable to, a uniform martingale on a two-letter alphabet. Austin's proof of the weak Pinsker property (2018), combined with Kalikow's Theorem 1.7(b), settles question (3) affirmatively for positive-entropy ergodic transformations. The remaining core — questions (1), (2), (4), and, for question (3), the zero-entropy case and the non-ergodic case not covered by Austin's theorem — remains open.

Background

A measure-preserving transformation $T$ on a probability space $(X,\mu)$, together with a finite or countable partition of $X$ into 'letters', defines a stationary process: a bi-infinite sequence of symbols recording which partition cell the orbit of a random point visits at each time step. When the partition generates (the cell containing a point is determined by its entire symbolic history), the process determines $T$ up to isomorphism. The Kolmogorov–Sinai entropy of a process measures its asymptotic rate of information production per time step; a Bernoulli shift is the process of independent, identically distributed symbols, and Ornstein's isomorphism theory shows that Bernoulli shifts of the same entropy are isomorphic.

Keane (1972) introduced g-measures: stationary processes for which the conditional probability of the present symbol given the entire infinite past is specified by a continuous function $g$ of the past (a 'g-function'). Kalikow (1990) singled out the subclass now called uniform martingales: processes for which the conditional distribution of the present symbol given only the $n$-step past converges, as $n\to\infty$, to the infinite-past conditional distribution uniformly over all possible pasts, in total variation distance. Equivalently, this is a strong, quantitative form of continuity of the g-function. 'Random Markov processes' are the general framework for such processes on countable alphabets studied in that line of work.

Kalikow's 'Infinite partitions and Rokhlin towers' (2012) proved two realization theorems (its Theorem 1.7): (a) every non-periodic measure-preserving transformation can be extended to a uniform martingale on a countable alphabet, where an extension is a uniform martingale process together with a factor map back onto the original transformation; and (b) if the transformation is a product of another transformation with a nontrivial Bernoulli shift, then it is in fact isomorphic to a countable-alphabet uniform martingale. The constructions rest on combinatorics of Rokhlin towers — finite stacks of disjoint measurable sets whose levels the transformation cycles through, a standard device for organizing the space when building symbolic codings. The same paper also proves a negative tower result (its Theorem 1.4): for a fair coin-flip event $A$, there is no subset $B\subset A$ that is a Rokhlin tower base whose associated error set has measure below $\mu(B)/6$, indicating sharp limits on how clean such tower constructions can be.

A transformation has the weak Pinsker property if for every $\varepsilon>0$ it is isomorphic to a direct product of a Bernoulli shift and a transformation of entropy less than $\varepsilon$. Thouvenot asked whether every ergodic transformation has this property, and Austin (2018) proved that every ergodic measure-preserving transformation does. Since a zero-entropy transformation admits no nontrivial Bernoulli factor, Kalikow's Theorem 1.7(b) now applies to every positive-entropy ergodic transformation, but its mechanism is unavailable in zero entropy. Kalikow also observed that an aperiodic zero-entropy process can never be a uniform martingale on a finite alphabet, which is why alphabet size and the entropy of the letter distribution are the delicate parameters in the questions below.

Problem Statement

Kalikow (2012, Section 1, immediately after Theorem 1.7) poses the following four questions about representing measure-preserving transformations as uniform martingales.

1. (Finite-alphabet extension.) Can every transformation with a finite alphabet be extended to a uniform martingale with a finite alphabet? If a countable alphabet is needed, can the induced measure on the alphabet be made to have finite Shannon entropy?

2. (Finite-alphabet isomorphism.) Is every positive-entropy transformation with a finite alphabet isomorphic to a uniform martingale with a finite alphabet? Kalikow notes it is easy to see that an aperiodic zero-entropy process cannot be a uniform martingale on a finite alphabet. What about the special case in which the transformation is a product of a Bernoulli shift and another transformation?

3. (Removing weak Pinsker.) Is every transformation isomorphic to a uniform martingale on a countable alphabet even if it does not obey the weak Pinsker property? Kalikow's parenthetical — that it was not known whether any transformation fails weak Pinsker — has been resolved by Austin's theorem, which shows every ergodic transformation obeys weak Pinsker; hence the positive-entropy ergodic case of this question is settled affirmatively by combining Austin's theorem with Theorem 1.7(b). The substantive surviving cases are zero entropy, where no nontrivial Bernoulli factor exists, and non-ergodic transformations, which Austin's theorem does not cover.

4. (Two-letter threshold.) Suppose there is a uniform martingale on a three-letter alphabet with entropy less than $\log 2$. Must it be isomorphic to a uniform martingale on a two-letter alphabet? Can it at least be extended to a uniform martingale on a two-letter alphabet?

The open core of the problem set is Questions 1, 2, and 4 together with the zero-entropy and non-ergodic cases of Question 3; the four questions form one coherent objective, a realization theory of measure-preserving transformations as uniform martingales with minimal or controlled alphabets. These parts are posed jointly as one problem set.

References

1. Steven Kalikow, 'Infinite partitions and Rokhlin towers', Ergodic Theory and Dynamical Systems 32(2):707–738 (2012), DOI 10.1017/S0143385711000381, https://doi.org/10.1017/S0143385711000381 (preprint arXiv:1108.5721, https://arxiv.org/abs/1108.5721)
2. Tim Austin, 'Measure concentration and the weak Pinsker property', Publications Mathématiques de l'IHÉS 128:1–119 (2018), DOI 10.1007/s10240-018-0098-3, https://doi.org/10.1007/s10240-018-0098-3 (preprint arXiv:1705.00302, https://arxiv.org/abs/1705.00302)
3. Neal Bushaw, Karen Gunderson, Steven Kalikow, 'Random Markov processes for countable and uncountable alphabets', Israel Journal of Mathematics 216(1):181–214 (2016), DOI 10.1007/s11856-016-1407-5, https://doi.org/10.1007/s11856-016-1407-5 (preprint arXiv:1410.1457 under the title 'Random-step Markov processes', https://arxiv.org/abs/1410.1457)
4. Steve Kalikow, 'Random Markov processes and uniform martingales', Israel Journal of Mathematics 71(1):33–54 (1990), DOI 10.1007/BF02807249, https://doi.org/10.1007/BF02807249
5. Michael Keane, 'Strongly mixing g-measures', Inventiones Mathematicae 16:309–324 (1972), DOI 10.1007/BF01425715, https://doi.org/10.1007/BF01425715
6. Tim Austin, Lewis Bowen, Christopher Shriver, 'Algebraic dynamical systems from LDPC codes satisfy a strong negation of the weak Pinsker property', Journal of the Institute of Mathematics of Jussieu 24:2317–2385 (2025), DOI 10.1017/S147474802510100X, https://doi.org/10.1017/S147474802510100X (preprint arXiv:2312.17387, 2023, https://arxiv.org/abs/2312.17387)
7. Séverin Benzoni, 'Introduction to weak Pinsker filtrations', arXiv:2504.00681 (2025), https://arxiv.org/abs/2504.00681