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. (Lean)

Manuscript

A smooth three-torus diffeomorphism with simple Lebesgue spectrum

September 23, 2026 · source · cite as \cite{OAI:A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026}

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.

Result 144 · Dynamical systems and ergodic theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.