Global integrability of the SRB-entropy gradient flow

Establish global-in-time existence of the gradient flow generated by the SRB entropy functional on the Hilbert manifolds A^{H^k}(M) (transitive Anosov diffeomorphisms) and E^{H^k}(M) (expanding endomorphisms): for every initial map f in A^{H^k}(M) or E^{H^k}(M), show that the unique trajectory Φ_t(f) of the SRB-entropy gradient vector field exists for all t in (−∞, ∞).

Background

The paper proves that the SRB entropy functional is Fréchet differentiable on suitable Sobolev (Hilbert) manifolds of uniformly hyperbolic systems and that its gradient vector field is locally Lipschitz, implying local existence of the corresponding gradient flow. Motivated by thermodynamic principles and the Gallavotti-Cohen Chaotic Hypothesis, the authors consider whether this locally defined flow extends to all times for all initial maps within the Hilbert manifolds of transitive Anosov diffeomorphisms A{Hk}(M) and expanding endomorphisms E{Hk}(M).

This conjecture asks to upgrade local integrability to global integrability, ensuring that the SRB-entropy gradient flow does not blow up or exit the manifold and is complete for all t ∈ (−∞, ∞).

References

Conjecture 1: (Global integrability of the gradient flow) The gradient flow Φ_t( f) exists for all t ∈ (−∞, ∞) for every f ∈ A{Hk}(M) or E{Hk}(M).

— Lipschitz Continuity and Formulas of the Gradient Vector of the SRB Entropy Functional  (2509.18596 - Chen et al., 23 Sep 2025) in Section 4: Questions Arising from the Gallavotti-Cohen Chaotic Hypothesis

While the local existence of the gradient flow follows directly from the fact that the derivative of the SRB entropy with respect to the map is Lipschitz continuous, the global existence, i.e, starting from an initial system with the SRB entropy within the interval $(0, h_{top})$, whether the local orbit can be extended in both directions of time indefinitely is unknown except in the case when the manifold is $S1$.

— Behavior of the SRB Entropy Functional in Families of Hyperbolic Attractors and Expanding Maps  (2609.00601 - Jiang, 1 Sep 2026) in Section 3, subsection “Differentiability of the SRB entropy and the Gradient Flow”