The approximate periodicity conjecture for long-time Floquet properties

Let $H(t)$ be a time-dependent Hamiltonian with period $1$, let $U(0,t)$ be its time-evolution operator, and let $|\psi(t)\rangle=U(0,t)|\psi(0)\rangle$ be the state evolved from an initial state $|\psi(0)\rangle$. Long-time properties of $|\psi(t)\rangle$ are periodic with period $1$. This conjecture concerns the proposed emergence of period-one behavior at long times despite the generally nonperiodic trajectory in Hilbert space. It does not hold for all Floquet systems.