OpenAI research collection · 1 manuscript · Source checked 2026-10-19
Resolves the probability-preserving form of Banach's simple Lebesgue-spectrum problem within smooth dynamics. A smooth volume-preserving diffeomorphism of the standard-volume three-torus has simple Lebesgue spectrum on its entire complex mean-zero L^(2) space: the bilateral iterates of one real observable form an orthonormal basis of that space.
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We solve the probability-preserving form of Banach's simple Lebesgue-spectrum problem in smooth dynamics. We construct a C^(∞) diffeomorphism of the three-torus that preserves standard volume and whose Koopman operator has simple Lebesgue spectrum on the entire mean-zero L^(2) space.
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