Conjecture on atomic limits of outlier measures

Let an atomic ergodic measure mean an ergodic invariant probability measure supported on a finite orbit, and let convergence of measures be weak convergence. The conjecture concerns the sequence of atomic measures arising from the outlier orbits described in the paper.

Atomic-limit conjecture. There exists a sequence of atomic ergodic measures converging to a measure with a positive atomic part.

This conjecture is motivated by numerical experiments with increasingly long outlier orbits and would produce a limiting invariant measure that is not Lebesgue measure. The paper presents it as a hypothesis requiring a rigorous proof, and its status remains open.