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ERRATA CORRIGE: Intrinsic algebraic entropy

T0 review · 1 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The Logarithmic Law for intrinsic algebraic entropy is true, and this note replaces the flawed original proof with a correct one.

desk verdict The counterexample to the original lemma is valid and valuable, but the replacement proof of the Logarithmic Law relies on a false claim lifted from DGSV, so the correction does not close the gap. read the letter →

arxiv 1908.09544 v1 pith:JSOSA6SX submitted 2019-08-26 math.GR math.ACmath.DSmath.RA

Daniele Toller , Simone Virili This is my paper · ORCID
classification math.GRmath.ACmath.DSmath.RA MSC 20K3020K2720K1522B0516D1037A3511R06
keywords intrinsicalgebraicentropyLogarithmicLawendomorphismsAbeliangroupsdynamicserratacorrigecounterexampleBernoullishift
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 is an erratum that repairs the proof of the Logarithmic Law for the intrinsic algebraic entropy $\widetilde{\mathrm{ent}}(\varphi)$ of an endomorphism $\varphi$ of an Abelian group. The law states that iterating the endomorphism $k$ times multiplies the entropy by exactly $k$: $\widetilde{\mathrm{ent}}(\varphi^k)=k\,\widetilde{\mathrm{ent}}(\varphi)$. The authors show that the original proof relied on a lemma whose second part is false, and they provide an explicit counterexample to that lemma. They then prove the law through a different route, using trajectory equalities, a reduction to finitely generated subgroups, and upper-continuity. The point that matters is that a basic property one expects from any entropy function is confirmed for the intrinsic algebraic entropy despite the flaw in its original demonstration.

What carries the argument

The machinery is the trajectory calculus of the intrinsic algebraic entropy. For a subgroup $H$, the $n$-th partial $\varphi$-trajectory is $T_n(\varphi,H)=H+\varphi(H)+\cdots+\varphi^{n-1}(H)$, and the entropy with respect to $H$ is the logarithmic growth rate of $\lvert T_n(\varphi,H)/H\rvert$ as $n\to\infty$; the intrinsic entropy $\widetilde{\mathrm{ent}}(\varphi)$ is the supremum of these growth rates over all $\varphi$-inert subgroups $H$. The central identity in the corrected proof is $T_n(\varphi^k,H)=T_{kn-k+1}(\varphi,H)$, which reindexes the $k$-fold iteration as a single trajectory and produces the factor $k$ in the growth rate. Around this identity, the argument uses Lemma 1(a) to find a finitely generated $H$ whose trajectory is all of $G$, Lemma 2.7(b) to transfer inertness from $\varphi$ to $\varphi^k$, Proposition 3.16(b) to identify the global entropy with the entropy on $H$, and Lemma 3.14 (upper-continuity) to extend from finitely generated trajectories to arbitrary $G$. The counterexample to Lemma 1(b) is a right Bernoulli shift on a direct sum of copies of $\mathbb{Z}(2)$, where two different inert subgroups give different partial entropies for $\beta^2$.

What would settle it

A concrete way to test the central claim is to compute $\widetilde{\mathrm{ent}}(\varphi^2)$ and $2\,\widetilde{\mathrm{ent}}(\varphi)$ for an explicit family of endomorphisms where the supremum over inert subgroups can be examined, for instance the right Bernoulli shift on the direct sum of countably many copies of $\mathbb{Z}(2)$; the note itself shows that one inert subgroup gives $\widetilde{\mathrm{ent}}(\beta^2,H)=\log(2)$ while $H'=T_2(\beta,H)$ gives $2\log(2)$, so what must be checked is whether the global suprema still coincide. Alternatively, a direct failure of Lemma 1(a), witnessed by a $\varphi$-inert subgroup $H$ with $\widetilde{\mathrm{ent}}(\varphi,H)\neq \widetilde{\mathrm{ent}}(\varphi,T_k(\varphi,H))$, would break the new proof even if the Logarithmic Law itself might still be true.

Watch

Extended reading notes

Core claim

The central claim is that the Logarithmic Law, $\widetilde{\mathrm{ent}}(\varphi^k)=k\,\widetilde{\mathrm{ent}}(\varphi)$, holds for every endomorphism $\varphi$ of an Abelian group and every positive integer $k$, even though the original argument for it was invalid. The note exhibits a right Bernoulli shift on the direct sum of countably many copies of $\mathbb{Z}(2)$ for which the equality asserted in Lemma 1(b) of the original paper fails, thereby showing that the old proof of the inequality $\widetilde{\mathrm{ent}}(\varphi^k)\le k\,\widetilde{\mathrm{ent}}(\varphi)$ cannot stand. A new proof reconstructs the missing inequality without the false lemma: for a finitely generated subgroup $F$ with $G=T(\varphi,F)$, the subgroup $H=T_{m+k-1}(\varphi,F)$ is both $\varphi$-inert and $\varphi^k$-inert, the partial trajectories satisfy $T_n(\varphi^k,H)=T_{kn-k+1}(\varphi,H)$, and the general case follows by taking suprema over finitely generated $\varphi$-invariant subgroups. The theorem originally stated in the paper being corrected is therefore true, with a proof that no longer depends on Lemma 1(b).

Load-bearing premise

The corrected proof assumes that Lemma 1(a) from the original paper remains valid even though the neighboring part 1(b) is false; if Lemma 1(a) also fails, the construction of the subgroup $H$ whose trajectories equalize $\varphi$ and $\varphi^k$ would not go through and the proof would have to be repaired again.

Editorial extensions

If this is right

  • For any endomorphism of an Abelian group, the entropy of the $k$-th iterate is exactly $k$ times the entropy of the endomorphism, so the invariant measures trajectory growth at the expected logarithmic rate.
  • The corrected proof does not disturb the other intrinsic entropy properties—additivity, upper-continuity, and the Intrinsic Yuzvinski Formula—because their proofs do not rely on the false Lemma 1(b).
  • The explicit counterexample to Lemma 1(b) warns that passing a subgroup to its trajectory closure can change the entropy contribution, so any future proof using that lemma must check the equality independently.
  • The identity $T_n(\varphi^k,H)=T_{kn-k+1}(\varphi,H)$ gives a direct computational shortcut for the entropy of iterates whenever a finitely generated subgroup generates the whole trajectory.
  • The reduction to finitely generated $\varphi$-invariant subgroups via upper-continuity means intrinsic entropy can be computed from the directed system of finitely generated trajectory subgroups.

Reading between the lines

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

  • The new proof leans on Lemma 1(a) without re-verifying it; a separate proof of that part, independent of the flawed Lemma 1(b), would make the corrected argument fully self-contained.
  • The Bernoulli-shift example suggests a general caution: for a fixed endomorphism, different inert subgroups can yield very different partial entropies, so only the supremum over all inert subgroups is a reliable invariant.
  • One testable extension is to carry the trajectory-reindexing argument to other entropy-like invariants defined by suprema over inert subgroups in settings that admit a direct-limit structure, where the same $k$-factor formula might appear.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 3 minor

Summary. This note is an erratum to the paper by Dikranjan, Giordano Bruno, Salce, and Virili on intrinsic algebraic entropy. The authors identify a false statement in the original Lemma 3.11(b), provide an explicit counterexample involving the right Bernoulli shift on a direct sum of copies of Z(2), and then propose a new proof of the Logarithmic Law, ~ent(φ^k)=k·~ent(φ), for endomorphisms of Abelian groups. The new proof first treats the case G=T(φ,F) with F finitely generated, using a cited result [DGSV, Prop. 5.6] to obtain an inert subgroup, and then extends to arbitrary G by a direct-limit argument and upper continuity.

Significance. If the proposed proof were valid, the note would be a valuable correction: the counterexample to Lemma 1(b) is explicit, self-contained, and correctly computed, with the two entropy limits (2 log 2 and log 2) genuinely disagreeing. The note is also transparent about the source of the original error. However, the new proof relies on a citation to [DGSV, Prop. 5.6] that is not justified in the setting used and is in fact false as applied. This is a load-bearing gap, because the finite-generation case is the core of the argument. The Logarithmic Law itself is likely true, and the gap appears repairable, but the manuscript as submitted does not provide a complete correct proof.

major comments (1)
  1. [Section 1, finite-generation case] The proof invokes [DGSV, Prop. 5.6] to assert the existence of m∈N+ such that T_m(φ,F) is φ-inert. This assertion is false in the exact setting where it is used. Let G=⊕_{i∈N}Z, let β be the right shift β(x_0,x_1,...)=(0,x_0,x_1,...), and let F=Z e_0. Then G=T(β,F), and ~ent(β)=0 because every β-inert subgroup H satisfies βH⊆H and hence has zero trajectory entropy. Yet for every m, T_m(β,F)=⊕_{i<m}Z e_i and (T_m+βT_m)/T_m ≅ Z e_m, which is infinite, so T_m is not β-inert. Thus the construction of H fails in this case, and the finite-generation part of the proof is invalid. The authors must either state and prove the precise version of Prop. 5.6 they need, with all hypotheses, or replace this step with a different argument.
minor comments (3)
  1. [Section 1, use of Lemma 1(a)] The proof cites [DGSV, Lem. 3.11(a)] without proof. Since part (b) of that lemma is false, the note should explicitly verify part (a) or cite an independent proof. This is not a substantive obstacle—part (a) follows because H'/H is finite and T_n(φ,H')=T_{n+k-1}(φ,H)—but the independence should be stated for the reader.
  2. [General case, Eq. (III)] The displayed formula for ~ent(φ^k) as a supremum over the subgroups T(φ,F) needs justification. Upper continuity gives a supremum over T(φ^k,F), not directly over T(φ,F). The equality can be recovered using T(φ,F)=T(φ^k,T_k(φ,F)) and the inclusion T(φ^k,F)⊆T(φ,F), but the text should spell this out.
  3. [Example 2] The notation Z(2) for the cyclic group of order 2 is nonstandard; Z/2Z or F_2 would be clearer.

Circularity Check

0 steps flagged · score 2.0 of 10

No exhibited circular reduction; the corrected proof imports self-authored DGSV lemmas, but the Logarithmic Law is not an input by construction.

full rationale

The new proof of the Logarithmic Law is not circular in the exhibited sense: the target identity ~ent(phi^k)=k·~ent(phi) is obtained from the definition of relative entropy, the trajectory equality T_n(phi^k,H)=T_{kn-k+1}(phi,H) in the constructed setting, and limits, after importing [DGSV, Prop. 5.6, Lem. 2.7(b), Prop. 3.16(b), Lem. 3.14] as external lemmas. None of these citations is a restatement of the Logarithmic Law, and the note explicitly asserts that their proofs do not rely on the false Lemma 1(b). The use of Lemma 1(a) from the same Lemma 3.11 is a possible missing-support/correctness issue, but the note does not define the target in terms of that lemma or exhibit an equation making the derivation self-referential; part (a) is a separate elementary statement, and its truth is not shown here to depend on the false part (b). The counterexample to Lemma 1(b) is self-contained and independent. The only mild burden is that the load-bearing lemmas come from a paper co-authored by one of the present authors; that is ordinary self-citation in an erratum and does not reduce the central claim to its own input. The skeptical counterexample to Prop 5.6, if valid, would be a defect in a cited theorem rather than circularity and cannot be adjudicated from this text.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central proof is a chain of reductions using published theorems from the same research group; none are proved in this note. The only genuinely new mathematical content is the counterexample and the identity T_n(phi^k,H)=T_{kn-k+1}(phi,H).

assumptions (5)
  • domain assumption [DGSV, Prop. 5.6]: for G=T(phi,F) with F finitely generated, there is m with T_m(phi,F) phi-inert
    Invoked without proof in the first part of the proof; its validity is not shown in this note and its proof is stated not to rely on Lemma 1(b).
  • domain assumption [DGSV, Lem. 2.7(b)]: H is phi^k-inert when H is phi-inert
    Used to infer that the finitely generated H is phi^k-inert.
  • domain assumption [DGSV, Prop. 3.16(b)]: if G=T(phi,H) and H is phi-inert, then ~ent(phi)=~ent(phi,H)
    Used to equate intrinsic entropy of the whole group with that computed with respect to H.
  • domain assumption [DGSV, Lem. 3.14]: upper continuity of ~ent under direct limits
    Used to reduce the general case to subgroups T(phi,F) with F finitely generated.
  • domain assumption [DGSV, Lem. 3.11(a)]: for H phi-inert and H'=T_k(phi,H), H' is phi-inert and ~ent(phi,H)=~ent(phi,H')
    The proof invokes this lemma without re-deriving it and without stating whether its proof uses the false Lemma 1(b); this is the weakest assumption.

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Pith. "Pith review of ERRATA CORRIGE: Intrinsic algebraic entropy." pith.science (2026). https://pith.science/paper/JSOSA6SX

@misc{pith2026190809544,
  author       = {Pith},
  title        = {Pith review of: ERRATA CORRIGE: Intrinsic algebraic entropy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JSOSA6SX}},
  note         = {Machine review of arXiv:1908.09544}
}
read the original abstract

The notion of intrinsic algebraic entropy of an endomorphism of a given Abelian group has been recently introduced in [D. Dikranjan, A. Giordano Bruno, L. Salce, S. Virili, Intrinsic algebraic entropy, J. Pure Appl. Algebra 219 (2015) 2933-2961]. In this short note we provide a correct argument to prove one of the basic properties of the intrinsic algebraic entropy: the Logarithmic Law. In fact, this property was correctly stated in [op. cit.] but, as we will show with an explicit counterexample, the original proof contains a flaw.

Discussion (0). Continue with ORCID to comment.

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