{"id":"openai-math/A-finite-entropy-system-without-a-smooth-positive-volume-model","pdf":"https://github.com/openai/math/blob/main/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026","tags":["ergodic-theory","dynamical-systems","smooth-dynamics"],"score":0,"title":"A zero-entropy system without a smooth positive-volume model","domain":"Dynamical systems and ergodic theory","family":"152","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.DS","summary":"We construct an ergodic invertible transformation of a standard nonatomic probability space with zero Kolmogorov–Sinai entropy that has no smooth positive-volume model. More precisely, it is not measurably conjugate to any C ∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. A single example excludes every finite dimension, including models on nonorientable manifolds and manifolds with smooth boundary.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-25","reactions":{},"formalized":false,"family_title":"Zero entropy does not guarantee a smooth positive-volume model"}