The ergodic-versus-geometric basin conjecture for hyperbolic cu-Gibbs states

Let $f$ be a smooth diffeomorphism with a hyperbolic $cu$-Gibbs state. Its geometric basin is the set of points whose empirical measures accumulate on that state, while its ergodic basin is the set of points whose empirical measures converge to it. Ergodic-versus-geometric basin conjecture. There are smooth diffeomorphisms with hyperbolic $cu$-Gibbs states whose ergodic basins are essentially different from their geometric basins. The conjecture concerns the possible failure of the equality, up to the relevant measure-theoretic notion of essentiality, between these two basin concepts; the paper notes that the known relation was obtained as a consequence of a GMY structure.