Topological and control theoretic properties of Hamilton-Jacobi equations via Lax-Oleinik commutators

In the context of weak KAM theory, we discuss the commutators $\{T_t^{-}\circ T_t^{+}{\}}_{t⩾0}$ and $\{T_t^{+}\circ T_t^{-}{\}}_{t⩾0}$ of Lax-Oleinik operators. We characterize the relation $T_t^{-}\circ T_t^{+}=Id$ for both small time and arbitrary time $t$. We show this relation characterizes controllability for evolutionary Hamilton-Jacobi equation. Based on our previous work on the cut locus of viscosity solution, we refine our analysis of the cut time function $\tau$ in terms of commutators $T_t^{+}\circ T_t^{-}-T_t^{+}\circ T_t^{-}$ and clarify the structure of the super/sub-level set of the cut time function $\tau$.