The Shannon-McMillan-Breiman theorem of random dynamical systems for amenable group actions

Bin Zhu; Yuan Lian

REVIEW 3 major objections 8 minor 15 references

The Shannon-McMillan-Breiman theorem of random dynamical systems for amenable group actions

T0 review · 3 major / 8 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The normalized information function of a finite partition converges in $L^1$ for continuous random dynamical systems over amenable groups, and almost surely to the fiber entropy when the invariant measure is ergodic.

desk verdict Plausible extension of SMB to random dynamical systems over amenable groups, but the lower-bound half of the main proposition is not proved and the novelty relative to earlier work is never established. read the letter →

arxiv 2506.01805 v1 pith:GKWBAGER submitted 2025-06-02 math.DS

Yuan Lian , Bin Zhu This is my paper

classification math.DS MSC 37A3537B40

keywords Shannon-McMillan-BreimantheoremamenablegrouprandomdynamicalsystemfiberentropytemperedFølnersequencepointwiseergodictheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to prove a Shannon–McMillan–Breiman theorem for random dynamical systems whose time evolution is indexed by a countable discrete amenable group rather than by the integers. It claims that for a continuous random dynamical system with an invariant measure, the normalized information function of a finite partition converges in $L^1$ to the conditional expectation of the fiber entropy production rate, and that under ergodicity the same quantity converges almost surely to the fiber entropy $h^{(r)}_\mu(\mathcal{F},\xi)$. A sympathetic reader would care because the result extends a basic information-theoretic limit to a broad class of group actions, making entropy of a random system a quantity visible on individual almost-sure trajectories. The argument decomposes the information function along a tempered Følner sequence and imports two covering lemmas from the pointwise ergodic theorem for amenable groups.

What carries the argument

The carrying object is the fibered information function $$I_{\mu_\omega}\left(\bigvee_{g\in F_n}$F^{{-1}}$_{g,\omega}\xi_{g\omega}\right)(x)=-\log\mu_\omega\left(\$xi^{{(F_n)}}$_\omega(x)\right),$$ where $\xi^{(F_n)}_\omega(x)$ is the atom of the refined partition containing $x$. The proof telescopes this quantity into a sum of conditional information terms along an enumeration of the Følner set, using the skew-product maps $\Theta_g(\omega,x)=(g\omega,F_{g,\omega}x)$ and the equivariance $F_{g,\omega}\mu_\omega=\mu_{g\omega}$. The martingale convergence theorem identifies the limiting information rate $r$, the pointwise ergodic theorem for amenable groups averages the terms, and two covering lemmas quoted from [9] bound the atom counts of the refined partition. The upper bound follows from a Borel–Cantelli estimate on a set of exceptional trajectories; the lower bound follows by showing that too many good atoms would contradict the definition of fiber entropy.

What would settle it

Find an ergodic continuous random dynamical system over a countable amenable group with a tempered Følner sequence and a finite partition for which the normalized information function has liminf strictly below the fiber entropy on a set of positive measure; a concrete place to look is a Bernoulli-type system over $\mathbb{Z}^2$ with independent random fiber maps, where the empirical normalized information along squares can be compared with the fiber entropy computed from the invariant measure.

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Extended reading notes

Core claim

The central claim is Proposition 3.1. For an infinite countable discrete amenable group $G$, a continuous random dynamical system $\mathcal{F}=\{F_{g,\omega}\}$ over a compact metric space, and a $G$-invariant measure $\mu$ with fiber measures $\mu_\omega$, every finite measurable partition $\xi$ of $\Omega\times X$ satisfies $$\frac{1}{|F_n|}I_{\mu_\omega}\left(\bigvee_{g\in F_n}$F^{{-1}}$_{g,\omega}\xi_{g\omega}\right)(x)\to E_\mu(r\mid\mathcal{I})(\omega,x)$$ in $L^1(\Omega\times X,\mu)$, where $r$ is the limiting martingale information rate and $\mathcal{I}$ is the invariant $\sigma$-algebra. If $\mu$ is ergodic, the convergence is almost everywhere and the limit is the fiber entropy $h^{(r)}_\mu(\mathcal{F},\xi)$. This is the random-dynamical-system analogue of the classical Shannon–McMillan–Breiman theorem for amenable group actions.

Load-bearing premise

The proof assumes that two covering estimates borrowed from an earlier theorem still work when the sets involved are built from the random maps and the group's large finite averaging sets; if these estimates fail, the lower bound in the main convergence result does not follow.

Editorial extensions

If this is right

In the ergodic case the normalized information function converges almost surely to the fiber entropy, so entropy can be read off from a single generic trajectory rather than from an ensemble average.
Without ergodicity the $L^1$ limit is the conditional expectation of the entropy production rate on the invariant $\sigma$-algebra, so each ergodic component carries its own rate.
The fiber entropy is attained as the limit of conditional entropies along a tempered Følner sequence, giving a concrete way to compute it from refining partitions.
The theorem applies to every infinite countable discrete amenable group, so the classical integer-action and multidimensional cases are recovered as special cases.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper does not state coding consequences, but the almost-sure convergence is exactly the ingredient a fiberwise noiseless coding theorem would need: a single code at rate $h^{(r)}_\mu(\mathcal{F},\xi)$ should compress almost every fiber realization of a random source over an amenable group.
Because the proof uses a tempered Følner sequence, a natural test is whether the almost-sure limit is independent of the choice of Følner sequence or whether non-tempered sequences can produce different pointwise rates.
The lower-bound step imports two covering lemmas from the deterministic pointwise ergodic theorem; checking their hypotheses for sets generated by random transformations would show whether the argument extends beyond the setting considered here.
If the same convergence held for refined partitions with shrinking diameters, it would imply a local entropy formula for random amenable-group actions, connecting fiber entropy to local information.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The paper sets out to give an SMB theorem for random dynamical systems over countable amenable groups. The honest one-line summary: the proof has a usable upper-bound half, but the lower-bound half of Proposition 3.1(3) is not actually proved, and the paper never makes clear which part of the theorem is new relative to Bogenschütz, Lindenstrauss, Ornstein-Weiss, and Weiss, all cited in the introduction.

What the paper does well: the telescoping computation in part (1) is clean and uses the cocycle property in the right way. The martingale step and the reduction to the amenable pointwise ergodic theorem are standard but not wrong. The upper-bound argument via Lemma 3.1 and Borel-Cantelli is also plausible. The authors are transparent that Lemma 3.1 and 3.2 are quoted from Lindenstrauss, which is honest.

The soft spot is where the theorem actually lives. The passage from Lemma 3.2 to the covering-number estimate for the set E′ is asserted in one sentence. Lemma 3.2 as quoted gives a measurable selection with certain cardinality and overlap properties, but nothing in the statement tells you how the µω-measure of the ξ^{F_n}-atom is controlled, nor how to count atoms selected for different (ω,x) without overcounting. That is exactly the step needed to contradict h^(r). The paper does not check whether Lindenstrauss's hypotheses—temperedness, the special form of the F_{i,j}, and the measure-class conditions—hold when the F_{i,j} come from a random dynamical system. So (3.4) is unsupported. There is also a smaller gap in part (1) where the passage from r_{i_n−m} to r under the averaged sum is hand-waved; likely fixable, but not written.

The novelty problem is just as serious. The introduction credits [1], [9], [11], and [13] with SMB theorems for random dynamical systems and amenable group actions. If those references cover this result, then the paper is a restatement with a less complete proof. If the authors believe their CRDS setting or their fiber entropy is a new case, they need to say so explicitly and prove that case. The citation pattern itself is honest, but it undercuts the claimed contribution.

Bottom line: the theorem is plausible but unverified. I would not cite it. I would still send it to a referee rather than desk reject, because the claim is concrete and a specialist could quickly determine whether the lower-bound gap is repairable. As submitted, it is not acceptable.

Referee Report

3 major / 8 minor

Summary. The paper claims to prove the Shannon-McMillan-Breiman theorem for continuous random dynamical systems over a countable discrete amenable group with a tempered Følner sequence. The main result is Proposition 3.1, which asserts L1 convergence of normalized information functions to E_μ(r|I), convergence of fiber entropy, and, under ergodicity, almost-sure convergence of normalized information functions to the fiber entropy. The proof decomposes the information function into a sum of conditional informations along the Følner set and invokes the pointwise ergodic theorem for amenable group actions; the lower bound for the ergodic case uses two combinatorial lemmas quoted from Lindenstrauss [9].

Significance. If correct, the result would be a natural extension of the Shannon-McMillan-Breiman theorem to random dynamical systems over amenable group actions, complementing prior work of Bogenschütz, Ornstein-Weiss, and Lindenstrauss. The statement is plausible and the overall strategy follows known templates, but the proof as written is incomplete at load-bearing points. The paper contains no numerical verification or machine-checked proofs; its contribution depends entirely on the validity of the analytic and combinatorial arguments, which are not fully supplied.

major comments (3)

[Section 3, proof of Proposition 3.1(3), lower bound (3.4)] The lower-bound half of Proposition 3.1(3) is not proved. After invoking Lemma 3.2, the manuscript asserts without proof that 'the number of elements covering E′ in ξ^{F_n} does not exceed 2^{(h^{(r)}−η)|F_n|}'. This step requires a counting or disjointness argument over atoms selected for different (ω,x), and it is precisely the step that produces the contradiction with h^{(r)}. Lemma 3.2 as quoted only gives set-valued random variables with cardinality properties; it says nothing about the μ_ω-measure of the ξ^{F_n}-atom containing x, nor does it supply a measurable selection in (ω,x). Without this argument, (3.4) does not follow and Proposition 3.1(3) is unproved.
[Section 3, proof of Proposition 3.1(1)] The telescoping identity expresses 1/|F_n| I_{μ_ω}(∨_{g∈F_n} F^{-1}_{g,ω} ξ_{gω})(x) as 1/|F_n| ∑_{m=1}^{i_n} r_{i_n−m}∘Θ_m(ω,x). To pass to the limit E_μ(r|I), one must justify replacing r_{i_n−m} by r inside the Følner average. The manuscript invokes the pointwise ergodic theorem and a 'control convergence theorem' after bounding sup_n r_n in L¹, but it does not show that 1/|F_n| ∑ |r_{i_n−m}−r|∘Θ_m tends to 0 μ-a.e. Pointwise convergence of r_n plus an L¹ bound on sup_n r_n is not by itself sufficient for this array-averaged convergence; an additional dominated-convergence or uniform-integrability argument is needed. This gap affects the proof of (1) and the derivation of (2).
[Section 3, Lemmas 3.1 and 3.2] Lemmas 3.1 and 3.2 are quoted verbatim from [9] and are used as the mechanism for both the upper and lower bounds in Proposition 3.1(3). The manuscript does not prove these lemmas, does not state their hypotheses in the random-dynamical-system setting, and does not verify those hypotheses for the sets F_{n,i} constructed from the tempered Følner sequence and the random maps F_{g,ω}. In particular, the compatibility of Lemmas 3.1–3.2 with the measurable decomposition μ = ∫ μ_ω dP is never checked. Since the lower bound depends on these lemmas, the main theorem is conditional on unverified external results.

minor comments (8)

[Section 2, Definition 2.1] The sentence 'd is a measure on X, m is a Borel probability measure' appears to confuse the metric d with a measure; this should be corrected.
[Section 2, Definition 2.1(1)] The statement that F_{e,ω} is a constant mapping is inconsistent with the cocycle identity; it should presumably be the identity map.
[Section 2, Definition 2.2] The phrase 'homomorphic maps' should be 'homeomorphisms'.
[Section 3, Lemma 3.1] The phrase 'there if B(i,a)' should read 'there exists B(i,a)'.
[Section 3, Lemma 3.2] The condition '1 = 1,2,...,M' should read 'i = 1,2,...,M'.
[Section 1, Introduction] Reference [3] has a stray line 'Lecture Notes in Mathematics, Spring-Verlag, Berlin/New York, 1983' that belongs to another entry.
[Throughout] Notation such as ξ^F_ω and F^{-1}_{g,ω} ξ_{gω} is used before being defined; a short notational paragraph would improve readability.
[Definition 2.3] The definition of h^{(r)} states a limit before proving that the limit exists; the proof of Proposition 3.1(2) supplies the convergence, but the definition should specify that the limit is taken along the given tempered Følner sequence.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the SMB limit is derived from the pointwise ergodic theorem and external combinatorial lemmas; the lower-bound gap is a proof concern, not a circular reduction.

full rationale

The derivation chain is not circular. The entropy h^{(r)}_mu(F, xi) is defined independently as a limit of normalized conditional entropies, and Proposition 3.1 derives the almost-sure convergence of normalized information functions from the pointwise ergodic theorem for amenable group actions, the martingale convergence theorem, and the external Lemmas 3.1 and 3.2 quoted from Lindenstrauss. The theorem is not used as its own input: the upper bound (3.2) follows from an entropy-covering set C(n), the Borel-Cantelli lemma, and the definition of h^{(r)}; the lower bound (3.4) is attempted by contradiction through Lemma 3.2 and a counting estimate. If the final counting estimate is asserted without proof, or if the hypotheses of Lindenstrauss's lemmas are not checked in the random dynamical system setting, these are genuine correctness or rigor gaps, but they are not instances of defining the conclusion in terms of itself, fitting a parameter and calling it a prediction, or importing a uniqueness theorem from the authors' own prior work. The only reference to prior work by one of the authors, [15], is contextual and not load-bearing for the proof. No step reduces by construction to its own inputs, so the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The proof rests on standard ergodic theory tools and on two combinatorial lemmas quoted from Lindenstrauss without verification. No free parameters or invented entities enter. The main unexamined premise is that the quoted lemmas apply to the random dynamical system's fibered partitions.

assumptions (4)

standard math For P-a.e. ω and all g∈G, F_{g,ω} µ_ω = µ_{gω} (equation 3.1).
Stated in the proof of Proposition 3.1 as following from the G-invariance of µ under the skew product and [1]; it is not proved in the paper.
standard math Theorem 4.28 of Kerr-Li [7], the pointwise ergodic theorem for amenable group actions, applies to the skew product (Ω×X, µ, G) and functions r ∈ L^1.
Used without proof in the proof of Proposition 3.1 to pass from pointwise sums over F_n to the conditional expectation Eµ(r|I).
domain assumption Lemmas 3.1 and 3.2 of Lindenstrauss [9] are valid for the particular Følner sets and random transformations appearing in this paper.
The paper quotes these lemmas directly and applies them to bound the cardinality of ξ^{F_n} atoms; their hypotheses are not verified in the random dynamical system setting.
standard math The sequence {F_n} is a tempered Følner sequence with e_G ∈ F_1 ⊆ F_2 ⊆ ... and |F_n| > n.
Assumed in Proposition 3.1; tempered Følner sequences exist for all countable amenable groups.

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Cite this review

Pith. "Pith review of The Shannon-McMillan-Breiman theorem of random dynamical systems for amenable group actions." pith.science (2026). https://pith.science/paper/GKWBAGER

@misc{pith2026250601805,
  author       = {Pith},
  title        = {Pith review of: The Shannon-McMillan-Breiman theorem of random dynamical systems for amenable group actions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GKWBAGER}},
  note         = {Machine review of arXiv:2506.01805}
}

read the original abstract

The Shannon-McMillan-Breiman theorem is one of the most important results in information theory, which can describe the random ergodic process, and its proof uses the famous Birkhoff ergodic theorem, so it can be seen that it plays a crucial role in ergodic theory. In this paper, the Shannon-McMillan-Breiman theorem in the random dynamical systems is proved from the perspective of an amenable group action, which provides a boost for the development of entropy theory in the random dynamical systems for amenable group actions.

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

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[2]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Entropy, pressure and a variational principle for random dynam- ical systems

Bogenschütz T. Entropy, pressure and a variational principle for random dynam- ical systems. Random Comput. Dynam, 1992, 1: 99–116

work page 1992

[2] [9]

Pointwise theorems for amenable groups

Lindenstrauss, E. Pointwise theorems for amenable groups. Invent Math, 2001, 146: 259–295

work page 2001

[3] [11]

The Shannon-McMillan-Breiman theorem for a class of amenable groups

Ornstein D S, Weiss B. The Shannon-McMillan-Breiman theorem for a class of amenable groups. Israel J Math, 1983, 44: 53–61

work page 1983

[4] [13]

Actions of amenable groups

Weiss B. Actions of amenable groups. In: Topics in Dynamics and Ergodic The- ory. London Mathematical Society Lecture Note Series, vol. 310. Cambridge: Cambridge University Press, 2003, 226–262

work page 2003

[5] [2]

The individual ergodic Theorem of information theory

Breiman L. The individual ergodic Theorem of information theory. Ann Math Stat, 1957, 28 (3): 809–811 Lecture Notes in Mathematics, Spring-Verlag, Berlin/New York, 1983

work page 1957

[6] [3]

A note on the ergodic theorem of information theory

Chung K. A note on the ergodic theorem of information theory. Ann Math Statist, 1961, 32: 612–614

work page 1961

[7] [4]

A mathematical theory of communication

Shannon C E. A mathematical theory of communication. Bell System Tech. J, 1948, 27 (4): 623–656

work page 1948

[8] [5]

Local entropy theory of a random dynamical system

Dooley A, Zhang G. Local entropy theory of a random dynamical system. Mem. Amer. Math. Soc., 2015, 233(1099)

work page 2015