Square Entropy and Uniform n-to-1 Bernoulli Transformations

Pouya Mehdipour; Somayeh Jangjooye Shaldehi

REVIEW 5 major objections 4 minor 20 references

Square Entropy and Uniform n-to-1 Bernoulli Transformations

T0 review · 5 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Square entropy is a complete isomorphism invariant for uniform n-to-1 Bernoulli transformations.

desk verdict A promising new entropy invariant, but the main classification theorem is false as stated and the proofs need serious repair. read the letter →

arxiv 2505.24647 v1 pith:QZBOF44T submitted 2025-05-30 math.DS math-phmath.MP

Pouya Mehdipour , Somayeh Jangjooye Shaldehi This is my paper

classification math.DSmath-phmath.MP MSC 37A0537A3537B40

keywords squareentropyzipshiftintrinsicergodicityBernoullitransformationextendedn-to-1mapvariationalprinciple

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

This paper introduces a new invariant, square entropy, for non-invertible measure-preserving maps. The invariant is the geometric mean of the forward and backward Kolmogorov-Sinai entropies. The authors claim that for uniform n-to-1 transformations of (m,l)-Bernoulli type—maps conjugate to full zip shifts—square entropy is a complete isomorphism invariant: two such maps with more than one 'past' symbol are isomorphic exactly when their square entropies agree. They also prove that full n-to-1 zip shifts are intrinsically ergodic with respect to square entropy, meaning a unique invariant measure maximizes it. If correct, this gives a classification result in a non-invertible setting where the classical Ornstein theorem does not apply.

What carries the argument

The central object is square entropy, h_{S,µ}(f) = √(h+µ(f) h-µ(f)), the geometric mean of the forward entropy computed from good image partitions (partitions whose atoms are forward invariant) and the usual backward Kolmogorov-Sinai entropy. For zip shifts the two factors are simply the entropies of the right-shift on the S-side and the left-shift on the Z-side, so the square entropy becomes √(ln l · ln m). A full zip shift is the coding that carries the argument: it represents an n-to-1 map by writing the future in S and the past in Z, joined by a surjection τ : S → Z; it is a local homeomorphism and expansive, and its invariant measures are exactly those whose marginals satisfy the consistency relation (1.9).

What would settle it

Find two uniform n-to-1 full zip shifts with the same (m,l) (hence the same square entropy) that are not measure-theoretically isomorphic; for instance, vary the map τ : S → Z so that the fiber sizes over Z differ, and check whether the resulting systems are still conjugate. If they are not, the classification fails; if they are, the missing isomorphism step needs a proof.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 4.5: among uniform n-to-1 transformations of (m,l)-Bernoulli type with #(Z)>1, the maps are measure-theoretically isomorphic if and only if they have the same square entropy, h_{S,µ}(f)=√(h+µ(f) h-µ(f)). The forward entropy h+ counts information along the future using a good image partition, and the backward entropy h- is the usual KS entropy. For a full zip shift with l right symbols and m left symbols, under the uniform measure these are ln l and ln m, so square entropy is √(ln m · ln l). Because for an n-to-1 map l = n m, equal square entropy pins down the pair (m,l), and the authors assert this forces isomorphism. They also establish the square variational principle and intrinsic ergodicity: the uniform product measure is the unique maximizer of square entropy for full zip shifts.

Load-bearing premise

The proof that equal square entropy implies isomorphism assumes, without proof, that any two uniform full zip shifts with the same numbers of left and right symbols are measure-theoretically isomorphic.

Editorial extensions

If this is right

If square entropy is unchanged by conjugacy, then equal square entropy is a necessary condition for isomorphism of uniform (m,l)-Bernoulli maps; Theorem 4.5 upgrades this to sufficiency.
Full n-to-1 zip shifts have a unique measure of maximal square entropy, namely the uniform measure, so they are intrinsically ergodic in the square-entropy sense.
The classical KS entropy alone cannot separate the two (2,4)-Bernoulli maps of Figure 1; square entropy assigns them different values, so it is a strictly finer invariant in this class.
The variational principle h_{S,top} = sup h_{S,µ} holds for n-to-1 zip shifts, giving a topological meaning to square entropy.

Reading between the lines

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

If the classification extends, square entropy could serve as an isomorphism invariant for larger classes of finite-to-1 endomorphisms, beyond the uniform zip-shift setting, wherever forward and backward entropy pairs are well defined.
The missing isomorphism step suggests a testable conjecture: all full zip shifts with the same (m,l) form a single measure-theoretic conjugacy class under uniform measures; this could be proven or refuted by constructing explicit conjugacies.
Square entropy may be related to the entropies of the natural extension and its quotient, since h- corresponds to the usual entropy and h+ to the factor entropy, which could give a geometric interpretation of the invariant.

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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,

Quick take: the square entropy object (geometric mean of forward and backward KS entropies) is a natural thing to look at for non-invertible maps, and the motivating example of two (2,4)-Bernoulli maps with equal KS entropy but different square entropy is a good illustration. But the main classification theorem is false as stated, and a couple of supporting claims don't survive contact with the definitions.

What's genuinely new: Section 2's definition of h_{S,µ} and h_{S,top}, and the idea that backward entropy on the quotient alphabet can be paired with forward entropy to separate examples like Figure 1. That is a reasonable research direction.

Soft spots, in order:

1. Theorem 4.5 fails as stated. The proof relies on 'the map ln m ln l with m,l > 1 is injective,' which is false: (2,64) and (4,8) both give sqrt(6) ln 2 as square entropy. The associated uniform full zip shifts have degrees 32 and 2 respectively, and degree is an isomorphism invariant. So the 'if' direction has a direct counterexample under the stated quantification, which does not fix n. If the intended theorem fixes n, that hypothesis is missing, and the proof would still need the unproven lemma that all uniform full zip shifts with the same (m,l) are isomorphic.

2. Proposition 1.18 claims expansivity using negative iterates of a non-invertible map. The definition of expansivity in the paper requires n ∈ Z, and σ_τ^{-i} is not defined as a map. So the proof does not establish the claim as written.

3. Theorem 3.5's uniqueness proof is skeletal. It shows the uniform measure attains the sup, but does not rule out other invariant measures achieving the same value of h_{S,µ}. The variational principle in Theorem 3.1 gives only an inequality; Theorem 3.2 asserts equality by pointing at the uniform measures, which isn't a substitute for a full argument.

4. Minor: the abstract says the characterization covers 'extended Bernoulli transformations,' but Theorem 4.5 is stated for uniform n-to-1 transformations with #(Z)>1; the connection between these classes is left vague.

Net: the paper has a salvageable idea and a fatal flaw. A fixed-n version might be repairable if the missing isomorphism lemma is true (it's not cited or proven). As is, I wouldn't cite it for the classification, but the square entropy definition might get a footnote in future work.

Recommendation for peer review: send it out, because the invariant is new and the counterexample is instructive; but the referee should catch the injectivity falsehood immediately and likely recommend rejection or major revision.

Referee Report

5 major / 4 minor

Summary. The paper defines a quantity called square entropy, h_S,mu(f)=sqrt(h+_mu(f) h-_mu(f)), for non-invertible measure-preserving maps, using a forward entropy associated to a good image partition and the usual backward/KS entropy. For full n-to-1 zip shift maps the authors compute h+(sigma_tau)=log l and h-(sigma_tau)=log m, so that square entropy is sqrt(log m log l). They prove a variational principle for square entropy, claim intrinsic ergodicity of n-to-1 full zip shifts with respect to square entropy, and state a classification theorem: uniform n-to-1 transformations of (m,l)-Bernoulli type with #(Z)>1 are isomorphic if and only if they have the same square entropy. The paper also introduces good image partitions and extended Bernoulli maps, and gives examples of (m,l)-Bernoulli transformations with equal KS entropy but different square entropy.

Significance. The idea of encoding the forward and backward branching of a non-invertible map by a product-type entropy is natural and potentially useful for classes of finite-to-one extensions of Bernoulli shifts. If the classification theorem were correct in a clearly fixed-degree setting, it would be a strong and interesting result: a one-number isomorphism invariant for a nontrivial class of non-invertible Bernoulli-type maps. The paper also makes a plausible and testable proposal for intrinsic ergodicity in this class. However, the central proof of the classification contains a false injectivity claim and a missing isomorphism step, and the definitions around the two entropy components are internally inconsistent. The results are therefore not reliable as written.

major comments (5)

[Theorem 4.5, p.16] The proof asserts that 'the map ln m ln l with m,l > 1 is injective.' This is false: sqrt(ln 2 * ln 64) = sqrt(ln 4 * ln 8), so equal square entropy does not by itself force equal (m,l). If the intended statement is for a fixed degree n, the proof should use the fact that for fixed n, l = n m and the function m -> ln m * ln(n m) is strictly increasing for m>1; the theorem statement must then say explicitly that n is fixed. Without fixing n, the 'if' direction has a direct counterexample with different degrees, since isomorphism preserves the cardinality of a.e. fibers.
[Theorem 4.5, p.16] Even after equal square entropy is repaired to give equal parameters (m,l), the proof jumps from m1=m2 to isomorphism without justification. One needs a lemma: any two full zip shift maps with the same balanced parameters (m,l) are measure-theoretically conjugate, for instance by a permutation of the alphabets that sends one fiber partition of tau to the other. This step is not stated or proved, and it is load-bearing for the classification claim.
[Theorems 3.2 and 3.5, pp.13-14] The variational equality h_S,top(sigma_tau) = sup_mu h_S,mu(sigma_tau) is asserted by writing sqrt(sup_muS h_muS(sigma_L) * sup_muZ h_muZ(sigma_R)) = sup_mu h_S,mu(sigma_tau). This equality is not automatic; the proof needs the explicit upper bound h_muS <= log l and h_muZ <= log m and the fact that the balanced uniform measure attains the product bound. The same gap affects Theorem 3.5, where uniqueness of the maximizing measure is claimed from Lemma 1.12 without showing that every maximizer of the product must maximize both factors. The claim is plausible and repairable, but the proof as written is incomplete.
[Section 2.1 and Lemma 2.6, pp.11-12] The definitions in Section 2 assign h+ to forward entropy using f^i of a good image partition (Eqs. (2.1)-(2.2)) and h- to the standard entropy using f^{-i} (Eq. (1.3)). The paragraph before Lemma 2.6 and the proof of the lemma instead identify h+ with the entropy generated by sigma_tau^{-i}(C_0), which is the standard entropy (log l), and h- with the entropy generated by sigma_tau^i(C_-1), which is the forward entropy (log m). This swaps the roles and contradicts the earlier definitions. The final formula (2.4), h_S = sqrt(log m log l), is clear, but the conceptual definition of the invariant is inconsistent and must be corrected.
[Proposition 1.18, pp.6-7] The claim that zip shift maps are expansive is false or ill-posed. Definition 1.10 uses f^n for n in Z although f is not invertible, and with the usual one-sided notion of expansivity the proof fails: two points differing only in coordinate -1 have distance 1/2 and their difference disappears after one forward iterate, so no delta > 0 separates them. This proposition is not used in the later arguments, but as a stated mathematical result it is incorrect and should be removed or replaced by a correct statement.

minor comments (4)

[Definition 1.10, p.5] The expansivity definition should explicitly require x != y; as written it also fails for x=y.
[Lemma 2.6, p.12] In the statement of Lemma 2.6, h-_mu(sigma_tau) = h_muS(sigma_R) should read h_muZ(sigma_R), since sigma_R acts on Sigma_Z.
[Throughout] There are several typos and stylistic issues: 'Howbeit' for 'However', 'deﬁed' for 'defined', 'hommeomorphism' for 'homeomorphism', 'suprimum' for 'supremum', and the notation in Definition 1.14 clashes with the use of m for #(Z). Please clarify the distinction between the number of partition elements in Definition 1.14 and the actual degree n = l/m of a zip shift map.
[Theorem 4.5, abstract] The abstract and Theorem 4.5 should state explicitly whether n is fixed before the classification claim; otherwise the statement is false, as noted in the major comments.

Circularity Check

1 steps flagged · score 4.0 of 10

Converse of Theorem 4.5 reduces square entropy to the (m,l) parameter that defines the class, then treats equal parameters as automatically isomorphic.

self definitional [Theorem 4.5 proof, Section 4; Definition 4.1; Lemma 2.6 and eq. (2.4)]
"Notice that once we have a n-to-1 zip shift map, #(S) = n(#(Z)). Indeed the map ln m ln l with m, l > 1 is injective. In other words, these entropies could be equal if and only if m1 = m2. This shows that uniform n-to-1 maps of (m,l)-Bernoulli type and equal square entropy are necessarily isomorphic."

Definition 4.1 defines the (m,l)-Bernoulli class as maps conjugate mod-0 to a full zip shift with parameters (m,l), and Lemma 2.6 with eq. (2.4) computes square entropy on that class as sqrt(ln l * ln m). Hence equality of square entropies reduces, by the paper's own formula, to equality of the parameter pair (m,l) (modulo the stated injectivity). The proof then jumps to 'necessarily isomorphic' without proving the structural fact that all uniform n-to-1 full zip shifts with the same (m,l) are mutually isomorphic. That unproved fact is precisely the content needed for the 'if' direction; without it the classification only unpacks the definition of the class, and with it the theorem is a corollary of the definition plus that independent fact.

full rationale

The paper is not wholly circular: square entropy is introduced independently, and the 'only if' direction of Theorem 4.5 is a legitimate invariance statement. The circularity is concentrated in the converse. For the class under study, Lemma 2.6 and eq. (2.4) fix h_S = sqrt(ln m * ln l), so equality of square entropy transparently reduces to equality of the parameter pair that already defines the (m,l)-Bernoulli class; the proof then assumes isomorphism follows from equal (m,l). Separately, the assertion that (m,l) -> sqrt(ln m * ln l) is injective is false as written (sqrt(ln 2 * ln 64) = sqrt(ln 4 * ln 8)), so the theorem as quantified is not correct; this is a correctness risk rather than a circularity. Self-citations to [MM]/[LM1] supply the zip-shift framework and definitions but are not the site of a uniqueness theorem that forces the conclusion. Score 4 reflects the independently defined invariant and genuine invariant direction against the definitional reduction in the converse.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

No data-fitting parameters appear. The free structural parameters m, l, and n are part of the class definition. The main unstated premise is the isomorphism of all zip shifts with fixed parameters, which makes the classification theorem close to a definitional unpacking.

assumptions (6)

standard math Variational principle for continuous maps on compact metric spaces.
Used to identify htop with sup of measure entropies for the two full shifts in Theorem 3.1 and 3.2.
standard math The full shift on a finite alphabet has a unique measure of maximal entropy, the uniform measure.
Lemma 1.12 is used to identify the unique maximizing measure for square entropy in Theorem 3.5.
domain assumption Invariant measures of a full zip shift are exactly the cylinder measures of the form (1.8) with PZ induced from PS by (1.9).
Stated as Remark 1.20 and used throughout Section 3, but no full proof is supplied in this paper.
domain assumption For an n-to-1 full zip shift, every fiber of tau has cardinality n, so #(S) = n #(Z).
Invoked in the proof of Theorem 4.5; it is true for uniform n-to-1 maps but is not stated as a definitional hypothesis.
ad hoc to paper All uniform full zip shifts with fixed (m,l) are mutually isomorphic.
Implicit in the final step of Theorem 4.5; never stated or proved, and it is load-bearing for the classification.
domain assumption For uniform zip shifts, h+ and h- equal log m and log l (up to order).
Lemma 2.6 is the stated proof, but the statement and proof contain swapped quantities and typos; the equality is plausible for the uniform case.

invented entities (1)

square entropy
purpose: Provide a new isomorphism invariant for non-invertible finite-to-1 maps.
Mathematical construction; its independent value rests entirely on the theorems proved in the paper, which are currently incomplete.

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Cite this review

Pith. "Pith review of Square Entropy and Uniform n-to-1 Bernoulli Transformations." pith.science (2026). https://pith.science/paper/QZBOF44T

@misc{pith2026250524647,
  author       = {Pith},
  title        = {Pith review of: Square Entropy and Uniform n-to-1 Bernoulli Transformations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QZBOF44T}},
  note         = {Machine review of arXiv:2505.24647}
}

abstract

In this paper, we define the so-called square entropy and prove that n-to-1 full zip shift maps are intrinsically ergodic. Furthermore, we show that square entropy characterizes uniform n-to-1 transformations of $(m,l)$-Bernoulli type that are extended Bernoulli transformations.

Figures

Figures reproduced from arXiv: 2505.24647 by the authors.

**Figure 1.** A 2-to-1 and a finite-to-1 Baker’s transformation. Both are (2, 4)-Bernoulli maps with the same Kolmogorov-Sinai (KS) entropy. Proof. Let στ represent a zip shift map on (m, l) symbols and take δ = 1/2. For any x, y ∈ Σ recall that ¯d(x, y) = 1 2M(x,y) where M(x, y) = ∞, if x = y min{|i|; xi 6= yi}, if x 6= y Let x 6= y. Then there exists some i ∈ Z such that xi 6= yi . Let i be the least in modulus of such i. If … view at source ↗

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

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Pith tools

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

[1] [1]

Syst.Theory 8, 193-202, 1974

Bowen, R., Some systems with unique equilibrium states, Math. Syst.Theory 8, 193-202, 1974

work page 1974

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Bowen, R., Bernoulli maps of the Interval, Israel Journal of Mathematics, Volume 28, issue 1-2, (1977)

work page 1977

[3] [3]

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work page 2009

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work page 2002

[5] [5]

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work page 2016

[6] [6]

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Shub, M.. Endomorphisms of compact differentiable manifolds. Amer. J. Math. 91 (1969), 171–199

work page 1969

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and Marcus, B., Symbolic Dynamics and Coding, Cambridge Univ

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work page Pith review arXiv doi:10.48550/arxiv.2502.11272