Uniqueness conjecture for Gibbs u-measures near the toral automorphism A

Let $A\colon \mathbb T^3\to\mathbb T^3$ be the hyperbolic automorphism induced by the displayed integral matrix, and let $f$ be an analytic diffeomorphism in a sufficiently small neighborhood of $A$. A Gibbs $u$-measure is an $f$-invariant measure whose conditional measures on strong unstable plaques are absolutely continuous with respect to the induced Riemannian volume on those plaques; an SRB measure is the Sinai–Ruelle–Bowen measure. Gibbs $u$-measure uniqueness conjecture. For all analytic diffeomorphisms $f$ in a sufficiently small neighborhood of $A$, there exists a unique Gibbs $u$-measure, which coincides with the SRB measure. The claim is motivated by numerical calculations suggesting weak convergence of the Gibbs $u$-measure averages and agreement with numerically computed SRB measures; the paper provides no proof or resolution.