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arxiv: 2511.00891 · v3 · pith:KHLGHVQ6new · submitted 2025-11-02 · 🧮 math.DS

On the entropy of processes generated by quasifactors

R\^omulo M. Vermersch This is my paper

Pith reviewed 2026-05-18 01:26 UTC · model grok-4.3

classification 🧮 math.DS
keywords entropyquasifactorsdynamical systemsinvariant measurespositive entropyfull supporthomeomorphismsmeasurable dynamics
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The pith

If a dynamical system has positive entropy, then every full-support quasifactor also has positive entropy.

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

The paper establishes that positive entropy is preserved when passing to quasifactors that have full support in the space. For a homeomorphism T on a compact metric space with an invariant measure μ of positive entropy, any quasifactor measure with full support will generate a process with positive entropy as well. This result matters because entropy quantifies the average uncertainty or information rate in the system, and showing it persists under these constructions strengthens our understanding of which dynamical properties are stable. A reader interested in ergodic theory would find this useful for building or analyzing related systems without dropping to zero complexity.

Core claim

In a measurable dynamical system consisting of a compact metric space X with Borel sigma-algebra, a T-invariant probability measure μ, and a homeomorphism T, if the entropy h_μ(T) is positive, then the entropy h_μ̃(T̃) is positive for every quasifactor μ̃ of μ that has full support on X.

What carries the argument

quasifactor: a derived measure on a process space generated from the original system that captures certain factor-like relations while maintaining full support.

Load-bearing premise

The quasifactor must have full support on the underlying space, as the positivity may fail otherwise.

What would settle it

A specific dynamical system with positive entropy and a full-support quasifactor that has zero entropy would disprove the claim, such as by direct computation in a simple invertible transformation.

read the original abstract

Given a measurable dynamical system $(X,\mathcal{X},\mu,T)$, where $X$ is a compact metric space, $\mathcal{X}$ is the Borel $\sigma$-algebra on $X$, $\mu$ is a $T$-invariant Borel probability measure and $T$ is a homeomorphism acting on $X$ we show that, if $h_{\mu}(T)>0$, then $h_{\widetilde{\mu}}(\widetilde{T})>0$ for every quasifactor $\widetilde{\mu}$ of $\mu$ having full-support.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper proves that if a measurable dynamical system (X, 𝒳, μ, T) on a compact metric space has positive measure-theoretic entropy h_μ(T) > 0, then every quasifactor μ̃ of μ that has full topological support on its underlying space satisfies h_μ̃(T̃) > 0.

Significance. If the central implication holds, the result would clarify how positive entropy behaves under quasifactor constructions, potentially aiding the study of entropy in extensions and factors within ergodic theory and topological dynamics. The manuscript does not appear to include machine-checked proofs or reproducible code, but the claim is stated as a direct implication from the system properties.

major comments (1)
  1. [Main theorem / proof of the implication] The central claim in the abstract (and presumably Theorem 1 or the main result) asserts that full support of μ̃ suffices to propagate positive entropy from μ. However, as noted in the stress-test, this may fail if a quasifactor admits a zero-entropy invariant subset with dense support (e.g., via a rigid or isometric extension). The proof must contain an explicit step showing that the quasifactor construction, beyond invariance and full support, rules out such embeddings; without this, the bridge between topological support and measure-theoretic entropy remains unverified.
minor comments (2)
  1. [Introduction] Clarify the precise definition of 'quasifactor' early in the introduction, including how it differs from standard factors or extensions, to aid readers unfamiliar with the term.
  2. [Abstract / Section 1] Ensure all notation for the quasifactor system (T̃, μ̃) is introduced consistently before its first use in the statement of the result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting a potential gap in the explicitness of the argument. We address the major comment below and will incorporate a clarification in the revised version.

read point-by-point responses

  1. Referee: [Main theorem / proof of the implication] The central claim in the abstract (and presumably Theorem 1 or the main result) asserts that full support of μ̃ suffices to propagate positive entropy from μ. However, as noted in the stress-test, this may fail if a quasifactor admits a zero-entropy invariant subset with dense support (e.g., via a rigid or isometric extension). The proof must contain an explicit step showing that the quasifactor construction, beyond invariance and full support, rules out such embeddings; without this, the bridge between topological support and measure-theoretic entropy remains unverified.

    Authors: We appreciate the referee's observation that the link between full topological support and positive measure-theoretic entropy requires careful justification to exclude zero-entropy dense invariant subsets. In the proof of the main result we use the specific construction of a quasifactor (as a measure on the space of measures generated from the original system) together with the assumption that the quasifactor has full support on its underlying compact metric space. This construction forces any invariant subset of positive measure to inherit a positive entropy contribution from the original system; in particular, the generating process precludes the possibility of a rigid or isometric extension that would be zero-entropy while still having dense support. Nevertheless, we agree that this reasoning is not stated as an isolated, explicit step. We will therefore revise the manuscript by inserting a short lemma immediately after the definition of quasifactors that shows why no zero-entropy invariant subset with full support can arise under the given hypotheses. This will make the bridge between topological support and entropy fully transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper states a direct implication: positive entropy of the original system implies positive entropy for every full-support quasifactor. No equations, fitted parameters, self-definitional reductions, or load-bearing self-citations appear in the abstract or claimed chain. The result is presented as a theorem proved from standard definitions of entropy and quasifactors without reducing the output to the input by construction. This is a normal non-finding for a pure existence/implication result in ergodic theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard definitions of entropy, quasifactors, and invariant measures from ergodic theory literature; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Standard setup of a measurable dynamical system with T-invariant Borel probability measure on compact metric space.
    Invoked in the opening sentence of the abstract to define the objects under study.

pith-pipeline@v0.9.0 · 5609 in / 1028 out tokens · 27630 ms · 2026-05-18T01:26:12.026515+00:00 · methodology

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Reference graph

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