Shub's entropy conjecture

Shub's entropy conjecture asks whether the topological entropy of a smooth self-map of a compact manifold is at least the logarithm of the spectral radius of its action on real homology, so that growth visible in homology forces exponential orbit complexity. A manuscript in OpenAI's public mathematics collection (github.com/openai/math), dated 25 September 2026, claims a counterexample to the general C¹ self-map formulation. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper or a formal proof is not presented here as independent verification.

Background

As recounted in the introduction of the manuscript:
- Shub related orbit complexity to homological growth in 1974. His later account explicitly includes endomorphisms as well as diffeomorphisms and distinguishes the earlier Lipschitz counterexamples discussed by Pugh from the C¹ question. In the general C¹ self-map formulation recorded by Saghin and Xia, the conjecture asks whether h_top(f) ≥ log ρ(f_*) for every C¹ map f of a compact smooth manifold without boundary, where ρ(f_*) is the spectral radius on the total real homology.
- Manning's inequality controls the action on first homology for every continuous map, and the C¹ lower bound of Misiurewicz and Przytycki concerns the topological degree, that is, top-dimensional homology. Yomdin proved the inequality for C^∞ maps by controlling homological growth through volume growth, and Gromov explained the regularity estimates.
- Among restricted C¹ classes, Saghin and Xia proved the bound for partially hyperbolic diffeomorphisms with one-dimensional center, and Liao, Viana and Yang for diffeomorphisms away from homoclinic tangencies.

The original result

Claim. Theorem 1.1 of A C¹ Counterexample to the Entropy Conjecture states that there are a positive integer q and a C¹ self-map f of the compact manifold (R/100Z) × (S²)^q+1 such that h_top(f) = 0 and the action of f on H₂ with real coefficients has eigenvalue 2, so that h_top(f) = 0 < log 2 ≤ log ρ(f_*). The manuscript states that this gives a negative answer to the general C¹ self-map formulation. The map is noninvertible and the eigenvalue lies in second homology of a manifold of dimension 2q + 3 ≥ 5, so the manuscript notes no conflict with the first-homology and top-dimensional positive results.
Method, as described: one sphere coordinate usually evolves by z ↦ z², a circle coordinate serves as a clock whose advance can become arbitrarily slow near one reading, and the remaining sphere coordinates are registers that store complex signals. During each clock passage, small signals are prepared for every possible later readout through an infinite prediction series whose first derivatives converge uniformly; the manuscript states that a finite truncation would not suffice. Every forward orbit approaches one of two compact invariant sets of zero entropy, and a localization lemma gives zero entropy without a uniform entrance time. Projection to the main sphere intertwines the homology action with that of z ↦ z², which the manuscript describes as a homological intertwining, not a dynamical semiconjugacy.
Source (openai/math repository, commit 7ef5c0c): A C1 Counterexample to the Entropy Conjecture, dated 25 September 2026.
Formalization, as described by the repository. The scope note lean/docs/151.md states that the formalization gives a counterexample to the general C¹ self-map version: a noninvertible C¹ map on a compact smooth manifold without boundary with topological entropy zero, whose action on second real homology has a nonzero eigenvector with eigenvalue 2. It links the comparator statement C1EntropyCounterexample.lean.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 28 October 2026)
2026-10-06	A C1 counterexample to the homological entropy bound, an article of the Latent Seminar blog at lowrank.github.io	Reading (exposition)	Explains the manuscript's claimed counterexample: topological entropy and homological growth, why a sphere-doubling map alone is insufficient, the infinite prediction series and its C¹ regularity, the orbit classification that gives zero entropy, and how the eigenvalue 2 on second homology is retained.	Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs.
2026-10-10	The entropy conjecture for C¹ maps, a course on Open Mathematics Courses (kokunoyumeto/open-math-courses-public)	Reading (exposition)	Four lessons: topological entropy and attraction, with the localization of entropy; a clock with prediction registers; the theorem that every orbit is attracted to one of two zero-entropy sets; and zero entropy together with the eigenvalue two coming from the degree of squaring.	The lessons state that they were written by Claude Opus 5.5 (Anthropic) in October 2026, one with a lemma and a coefficient comparison added by GPT-6 Astra (OpenAI), and self-checked by the writing AIs; the course lists its status as author self-checked.
2026-10-11	A counterexample to the Shub entropy conjecture, an expository note by Andrey Gogolev, Ohio State University	Extension (exposition)	For every finite r ≥ 1, constructs a noninvertible C^r self-map of S¹ × S² × S² × S² with zero topological entropy and eigenvalue 2 on second homology, in which every orbit converges to a fixed point or is asymptotic to an orbit of period 20. The note states that the two auxiliary spheres can be replaced by one S³, that finite-smoothness counterexamples likely also exist for diffeomorphisms, and that the conjecture for aspherical manifolds remains open.	Expository note dated 11 October 2026 on the author's university web page; not peer reviewed. The note credits the construction to the OpenAI manuscript, states that its author claims no credit for it and used an OpenAI model to simplify and understand the proof, and calls the argument elementary. Listed among follow-up works in the NOTES of the community revision mathcommunity/oai-math-revised. Checked at the source 28 October 2026.
2026-10-16	AI Proved What?, episode 151 Wrapping without chaos?, remade version of 26 October 2026	Reading (exposition)	Video explainer of the claim that there are a positive integer q and a C¹ self-map of a circle times q + 1 two-spheres with zero topological entropy whose action on second real homology has eigenvalue 2, so that the general C¹ self-map formulation of Shub's entropy conjecture fails. The episode states that the paper claims nothing about diffeomorphisms, homeomorphisms or maps smoother than C¹, where Yomdin's theorem rules out a C∞ example, and that it fixes no particular q.	Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as a preprint: OpenAI, 15 pages, dated September 25, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: the main theorem, the site has not run the check. The page also states that the formalization comprises 201 files and 22,878 lines of proof with no unproved steps that the site could find, that the Lean proves noninvertibility by showing that the map is not onto whereas the paper shows that it is not one-to-one, that OSVerify has not reviewed the paper and that the site has seen no expert review. An update of 27 October 2026 describes Gogolev's expository note and states that its C^r claim is the note's, not the paper's, and that the site has not checked it.

Notes

The background summarizes the manuscript's own account of earlier work.
The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.