Have you looked into “conditionally conserved” quantities/symmetries here? Most macroscopic properties fall into this category—e.g. the color of a particular material is conserved so long as it doesn’t change phase or transmute (i.e. it stays within a particular energy range). This is associated with a (spontaneously-broken) symmetry, since the absorption spectrum of a material can be uniquely determined from its space group. I’d be willing to bet that the only information accessible at a distance (up to a change of variables) are these conditionally conserved quantities, but I’ve had a hard time rigorously proving it. (I believe it requires KAM theory).
Also, have you looked into Koopman spectral theory? It’s for deterministic systems, but seems quite relevant—the Koopman operator maps observables at one time to observables at another, telling you how they change over time. You can relate its spectrum to underlying geometrical properties, and it’s a linear operator even if your dynamics are nonlinear.
Have you looked into “conditionally conserved” quantities/symmetries here? Most macroscopic properties fall into this category—e.g. the color of a particular material is conserved so long as it doesn’t change phase or transmute (i.e. it stays within a particular energy range). This is associated with a (spontaneously-broken) symmetry, since the absorption spectrum of a material can be uniquely determined from its space group. I’d be willing to bet that the only information accessible at a distance (up to a change of variables) are these conditionally conserved quantities, but I’ve had a hard time rigorously proving it. (I believe it requires KAM theory).
Also, have you looked into Koopman spectral theory? It’s for deterministic systems, but seems quite relevant—the Koopman operator maps observables at one time to observables at another, telling you how they change over time. You can relate its spectrum to underlying geometrical properties, and it’s a linear operator even if your dynamics are nonlinear.