OAI.ThreeTorus.simple_lebesgue_spectrum

The theorem states that on the 3-torus (ℝ/ℤ)³ with normalized product Haar (Lebesgue) probability measure μ, there exists a bijection T of the torus such that T and its inverse are both smooth (C^∞), in the sense that near every point of the universal cover ℝ³ they are locally given by a C^∞ lift, and T preserves μ. Moreover, there is a real-valued function f in the mean-zero subspace H₀ of L²(μ) (complex-valued, with zero integral; real means the imaginary part vanishes almost everywhere) and a family v_n, n ∈ ℤ, in H₀ with v_n = f∘Tⁿ almost everywhere, such that (v_n) is orthonormal and its complex linear span is dense in H₀. In addition, T is ergodic with respect to μ, and there is a linear isometric isomorphism W from H₀ onto L²(ℝ/ℤ) with Haar probability measure such that, for every g in H₀, the composition g∘T is again in H₀ (a class h with h = g∘T almost everywhere) and W(h)(z) = e^{2πiz}·W(g)(z) for almost every z. Thus the Koopman action of T on mean-zero functions is unitarily equivalent to multiplication by the first Fourier character on the circle L² space, which is the formal statement of simple Lebesgue spectrum. The proof is admitted in the source, not verified.