Infimum for nonsmooth SRB measures and Axiom A diffeomorphisms

Determine whether the SRB entropy infimum can be reduced to zero for diffeomorphisms preserving a nonsmooth SRB measure or for Axiom A diffeomorphisms whose topological pressure is negative, including verification of the required perturbation and distortion estimates.

Background

The paper proves that, for a path-connected component containing a diffeomorphism with a hyperbolic attractor, a suitable perturbation path can make the SRB entropy tend to zero. It then identifies two settings in which the argument is not established: preservation of an SRB measure without a continuous Lebesgue density, and Axiom A diffeomorphisms for which the relevant pressure is negative.

In the Axiom A case, the SRB entropy includes a pressure term and the absence of stable foliations may alter the distortion estimates needed to control the entropy integral. The unresolved issue is whether the same zero-infimum conclusion remains valid in these broader settings.

References

The infimum of the SRB entropy problem remains a project incomplete. We do not know whether the theorem holds in the cases of (1) $f$ preserves a non-smooth SRB measure, i.e., an SRB measure without a continuous density function with respect to the Lebesgue measure. (2) $f$ is an Axiom A diffeomorphism, where the topological pressure is negative.

— 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 1

If we impose the boundedness condition in $C2$ norm of $f_t$, we do not know whether the infimum is still zero.

— 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 2