Banach's simple Lebesgue-spectrum problem at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Banach's simple Lebesgue-spectrum problem

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	things that move forever	Levels:	1
Category:	Dynamical systems and ergodic theory	Lean version:	YES! ✔
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Banach’s simple Lebesgue-spectrum problem. 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 L² space: the bilateral iterates of one real observable form an orthonormal basis of that space.

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LEVEL 1 A smooth three-torus diffeomorphism with simple Lebesgue spectrum

released 2026-09-23 | 1 theorem · 8 lemmas · 12 proofs · 12,770 words | PLAY LEVEL 1 » (pdf)

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 L² space.

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