Long-time behavior of generalized gradient flows of solutions to Hamilton-Jacobi equations

We study the long-time behavior of the generalized gradient flow associated with solutions of the critical Hamilton-Jacobi equation for mechanical Hamiltonians on the flat torus. For any semiconcave function, we show that its critical set -- points whose superdifferential contains the zero vector -- acts as an approximate attractor for the flow. When the function is a solution of the critical equation, the critical set decomposes into regular and singular parts, and we establish a dichotomy describing which part trajectories approach as $t \to \infty$. Our analysis uses limiting occupational measures, a class of invariant measures capturing the asymptotic distribution of the flow. An essential ingredient is a complete proof of the global invariance of the singular set, a result previously announced by Albano (2016) but not fully established.