The entropy conjecture for C¹ maps

OpenAI's theorem (2026) that Shub's entropy conjecture fails for C^1 maps: a noninvertible C^1 self-map of a compact manifold with zero topological entropy and the eigenvalue 2 on second homology; through entropy localization, a clock with prediction registers, an attraction theorem and the degree of squaring.

Four lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Topological entropy and attraction
2. A clock with registers
3. Every orbit is attracted
4. Zero entropy and the eigenvalue two

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Local tools for bundles and transport (Bundles, curvature and holonomy)
• Cap products and cohomology with compact supports (Poincaré duality on manifolds)
• Orientations and fundamental classes (Poincaré duality on manifolds)
• Poincaré duality (Poincaré duality on manifolds)
• Metric spaces, compactness and differentiation in several variables (core courses Real Analysis I and II), singular homology with excision and homotopy invariance (core course Algebraic Topology), and smooth manifolds with cutoff functions.