Banach's problem: a probability-preserving transformation with simple Lebesgue spectrum

Theorem 1.1: there are a $C^\infty$ volume-preserving diffeomorphism $T$ of $\mathbb T^3$ and a real $f\in L^2_0(\mathbb T^3)$ such that $\{f\circ T^n\}_{n\in\mathbb Z}$ is an orthonormal basis of $L^2_0$; hence $T$ is ergodic and $U_T$ on $L^2_0$ is unitarily equivalent to multiplication by $w$ on $L^2(S^1)$. Mixing, zero entropy and zero Lyapunov exponents follow; mixing of all orders uses a separate multiple-mixing manuscript. The manuscript states this is distinct from the historical real-line question attributed to Banach and does not address that formulation; $f$ is only $L^2$, not smooth.

Theorem 1.1: there are a $C^\infty$ volume-preserving diffeomorphism $T$ of $\mathbb T^3$ and a real $f\in L^2_0(\mathbb T^3)$ such that $\{f\circ T^n\}_{n\in\mathbb Z}$ is an orthonormal basis of $L^2_0$; hence $T$ is ergodic and $U_T$ on $L^2_0$ is unitarily equivalent to multiplication by $w$ on $L^2(S^1)$. Mixing, zero entropy and zero Lyapunov exponents follow; mixing of all orders uses a separate multiple-mixing manuscript. The manuscript states this is distinct from the historical real-line question attributed to Banach and does not address that formulation; $f$ is only $L^2$, not smooth.