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Shub's entropy conjecture for C1 self-maps

For a continuous self-map ff of a compact manifold MM, let htop(f)h_{\mathrm{top}}(f) be its topological entropy and ρ(f∗)\rho(f_*) the spectral radius of the induced map on real homology H∗(M;R)H_*(M;\mathbb R). Shub (1974) conjectured that htop(f)≥log⁡ρ(f∗)h_{\mathrm{top}}(f)\ge\log\rho(f_*) for every C1C^1 map; Lipschitz counterexamples were known, Manning proved the bound for H1H_1, Misiurewicz-Przytycki for top degree, and Yomdin for C∞C^\infty maps, with further C1C^1 diffeomorphism cases (Saghin-Xia, Liao-Viana-Yang). Does htop(f)≥log⁡ρ(f∗)h_{\mathrm{top}}(f)\ge\log\rho(f_*) hold for every C1C^1 self-map ff of a compact smooth manifold without boundary?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Smooth dynamical systems; topological entropy and homology
Posed by
Michael Shub
Year posed
1974
Years open
52y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
48 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for some q≥1q\ge1 there is a C1C^1 self-map ff of M=(R/100Z)×(S2)q+1M=(\mathbb R/100\mathbb Z)\times(S^2)^{q+1} with htop(f)=0h_{\mathrm{top}}(f)=0 while f∗f_* on H2(M;R)H_2(M;\mathbb R) has eigenvalue 2, so htop(f)<log⁡ρ(f∗)h_{\mathrm{top}}(f)<\log\rho(f_*). The map is noninvertible, dim⁡M=2q+3≥5\dim M=2q+3\ge5, and the eigenvalue sits in intermediate degree, consistent with the known first-homology, top-degree and C∞C^\infty positive results. It does not address C1C^1 diffeomorphisms, CrC^r maps for r≥2r\ge2, or the conjecture in low dimensions.

What the AI did

The release README says every result in it was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The manuscript (September 25, 2026) has no companion in the release.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture in its general C1C^1 self-map form; it gives a noninvertible C1C^1 map of (R/100Z)×(S2)q+1(\mathbb R/100\mathbb Z)\times(S^2)^{q+1} with zero entropy and eigenvalue 2 on H2H_2, which contradicts that form. The map is not a diffeomorphism, so the C1C^1 diffeomorphism version is untouched, as the paper indicates. The release's lean/docs/151.md points to ComparatorChallenges/C1EntropyCounterexample.json, which is not in the formalization catalogue; its solution module OAI/Dynamics/EntropyCounterexample/Counterexample.lean exists at the pinned commit. The statement was read here: for some q>0q>0 a map of the unit circle times (S2)q+1(S^2)^{q+1} that is C1C^1 in the sense of local C1C^1 extensions in the ambient Euclidean space, with Mathlib cover entropy 0, a nonzero real singular H2H_2 class with eigenvalue 2, and not bijective. This states the headline. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.

Sources

Changelog1 change

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