Supremum of SRB entropy in hyperbolic components

Determine whether the supremum of the SRB entropy over the path-connected open component of a uniformly hyperbolic diffeomorphism always equals the corresponding topological entropy.

Background

The entropy of every invariant measure is bounded above by topological entropy, but equality for the supremum of the SRB entropy is not established in general. Equality is known in certain toral Anosov components containing an Anosov automorphism, while higher-dimensional components may contain no automorphisms.

The unresolved problem concerns whether the upper bound is nevertheless attainable as a supremum throughout arbitrary path-connected open components of uniformly hyperbolic systems.

References

However, whether the supremum of the SRB entropy in ${Cr}(f_1)$ is the same as the topological entropy, in general, is unknown.

— Behavior of the SRB Entropy Functional in Families of Hyperbolic Attractors and Expanding Maps  (2609.00601 - Jiang, 1 Sep 2026) in Section 2, Remarks, Remark 3

In general, it is unknown to the author, whether every Riemannian manifold that admits an expanding map also admits an expanding map whose SRB measure and the measure of maximal entropy coincide.

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