Result 151, Dynamical systems and ergodic theory

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.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

Can growth in the topology of a dynamical system force complicated motion? The manuscript reports a counterexample: repeated application of a map can amplify a two-dimensional homological feature while its orbit complexity has zero exponential growth rate.

What changes?

The construction is a noninvertible, continuously differentiable self-map of a compact smooth manifold without boundary. Its topological entropy, which measures the exponential growth rate of distinguishable orbit segments, is zero. Yet its action on second real homology, an algebraic description of two-dimensional cycles with real coefficients, has eigenvalue 2. The manuscript therefore claims to disprove the homological entropy lower bound in Shub's conjecture for general continuously differentiable self-maps, not for every more restrictive class of maps.

What does that help mathematicians do?

The eigenvalue 2 means that iteration doubles a nonzero homology class at each step. The conjectured bound would therefore require entropy at least the logarithm of 2, rather than zero. This gives researchers a specific obstruction: exponential growth detected by homology alone cannot guarantee positive topological entropy under these assumptions. Any theorem making that inference needs additional restrictions that exclude this counterexample.

Are there practical applications?

The immediate value is foundational for dynamical systems. Homology provides an algebraic way to track how a map transforms a space; entropy describes the complexity of its trajectories. This construction identifies a limit to using the former as a proxy for the latter, helping clarify which assumptions future entropy bounds must address.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A C^1 Counterexample to the Entropy Conjecture

September 25, 2026 15 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/151.md.

A C1 counterexample to the entropy conjecture

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Shub's entropy conjecture predicts that a smooth self-map's topological entropy is at least the logarithm of the spectral radius of its action on real homology. The formalization gives a counterexample to the general C1C^1 self-map version: a noninvertible C1C^1 map on a compact smooth manifold without boundary has topological entropy zero, while its action on second real homology has a nonzero eigenvector with eigenvalue 22. Thus the homological lower bound is strictly positive and fails for this map.

Comparator links

Result Comparator statement
A C1C^1 counterexample to the entropy conjecture C1EntropyCounterexample.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.