A C^{1} 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)

Manuscript

A C^1 Counterexample to the Entropy Conjecture

September 25, 2026 · source · cite as \cite{OAI:A-C1-Counterexample-to-the-Entropy-Conjecture-September-25-2026}

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.

Result 151 · Dynamical systems and ergodic theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.