Family 151Dynamical systems and ergodic theoryNo LeanNot yet digested
A C counterexample to the entropy conjecture
Constructs a noninvertible C 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 self-maps: homological growth need not force positive orbit complexity.
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A C^1 Counterexample to the Entropy Conjecture
Sep 25, 2026·article.pdf
We disprove the general C self-map formulation of Shub's entropy conjecture. We construct a noninvertible C 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.