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A C^1 Counterexample to the Entropy Conjecture
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Abstract
We disprove the general C1 self-map formulation of Shub's entropy conjecture. We construct a noninvertible C1 self-map of a compact smooth manifold without boundary whose topological entropy is zero, while its action on second real homology has eigenvalue 2. Thus the topological entropy is strictly smaller than the logarithm of the homological spectral radius.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope notes for family 151 (lean/docs/151.md, 2026). The release's scope note describes a Lean development of a noninvertible $C^1$ self-map with zero topological entropy and an eigenvector of eigenvalue 2 on second real homology; its linked comparator statement specifies the sphere-product manifold, nonbijectivity and dimension bound but ends with a
sorryproof placeholder.
Possibly relevant literature (found by the agent, not checked)
- D. Fried and M. Shub, Entropy, linearity and chain-recurrence (Publ. Math. IHÉS 50, 1979, pp. 203–214). Classic paper on the entropy conjecture and its low-regularity failures; background for the paper's statement that earlier counterexamples were Lipschitz and the $C^1$ question was distinct.
- I. Kangasniemi, Y. Okuyama, P. Pankka, T. Sahlsten, Entropy in uniformly quasiregular dynamics (arXiv:1903.10183, 2019). Proves Shub's entropy conjecture for non-injective uniformly quasiregular self-maps ($h(f)=\log\deg f$), a positive result for a class of noninvertible maps to compare with the paper's claim that noninvertibility escapes known positive results.
- S. Sandfeldt, Bounds of the spectral radius of the induced map on cohomology (arXiv:2106.07572, 2021). For $C^1$ diffeomorphisms with spectral radius $>1$ on cohomology it finds an ergodic measure with a positive Lyapunov exponent; relevant to the paper's remark that its counterexample is noninvertible and does not address $C^1$ diffeomorphisms.
Digests and expositions
- Open Mathematics Courses, The entropy conjecture for C¹ maps (October 2026). Four lessons develop entropy localization, the clock and registers, orbit attraction, and the homological eigenvalue calculation, with proofs and exercises; the site attributes the writing to Claude Opus 5.5 with additions by GPT-6 Astra and says independent review is incomplete. [AI-generated per source]
- Gradient Damage, Birkhoff Conjecture, Zero Entropy and 7 More in Dynamical Systems and Ergodic Theory (video, 15 October 2026). The chapter beginning at 1:46 introduces Shub's entropy conjecture and the claimed counterexample; the page describes the animations as illustrations rather than proofs.
News and blogs
- rmj.videotoblog.ai, Protipříklad třídy C¹ k entropické domněnce: topologie roste, ale entropie zůstává nulová (VideoToBlog, undated). This Czech exposition explains the clock, prediction registers and homological calculation, distinguishing the claimed noninvertible C¹ counterexample from the diffeomorphism and C∞ cases.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 151
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-C1-Counterexample-to-the-Entropy-Conjecture-September-25-2026,
author = {{OpenAI}},
title = {{A $C^1$ Counterexample to the Entropy Conjecture}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-C1-Counterexample-to-the-Entropy-Conjecture-September-25-2026/article.pdf}{OAI:A-C1-Counterexample-to-the-Entropy-Conjecture-September-25-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/151.md. The Lean statements have not been checked by people against the paper's claims.
Family 151: A C1 counterexample to the entropy conjecture
Constructs a noninvertible C1 self-map of a compact smooth manifold with zero topological entropy but eigenvalue 2 on second homology. This disproves the homological entropy lower bound for general C1 self-maps: homological growth need not force positive orbit complexity.
This is the only paper in this family.