OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
Constructs a noninvertible C^(1) 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 C^(1) self-maps: homological growth need not force positive orbit complexity.
Read the manuscripts
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
MANUSCRIPT 01
A C^1 Counterexample to the Entropy Conjecture
We disprove the general C^(1) self-map formulation of Shub's entropy conjecture. We construct a noninvertible C^(1) 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…
Full source abstract
We disprove the general C^(1) self-map formulation of Shub's entropy conjecture. We construct a noninvertible C^(1) 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.
This exact PDF is not listed in the imported formalization catalogue. Family scope notes may contain further information.
New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.