Existence of equilibrium along the SRB-entropy gradient flow

Determine whether trajectories of the SRB-entropy gradient flow on the Hilbert manifolds A^{H^k}(M) (transitive Anosov diffeomorphisms) and E^{H^k}(M) (expanding endomorphisms) converge as t → ∞: given any initial map f with SRB entropy (f) strictly less than the topological entropy h_top, prove that the limit lim_{t→∞} Φ_t(f) exists.

Background

Building on local existence of the SRB-entropy gradient flow and physical motivations from maximum entropy production and the Gallavotti-Cohen Chaotic Hypothesis, the authors formulate a convergence question toward equilibrium (a map with vanishing SRB-entropy gradient).

Within any path-connected component of A^{H^k}(M) or E^{H^k}(M), the SRB entropy is bounded above by the topological entropy. The conjecture posits convergence to an equilibrium whenever the initial SRB entropy is strictly below this bound, thus formalizing an analogue of thermodynamic relaxation to equilibrium in these dynamical families.