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 [Formula: see text]. 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.