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A zero-entropy system without a smooth positive-volume model

Source: github.com/openai/math/tree/main/preprints/A-finite-entropy…

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OpenAI Mathopenai-math/A-finite-entropy-system-without-a-smooth-positive-volume-modelSep 25, 2026math.DSAI-generatedscore 0

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.

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