No-wild-historic-behavior Lyapunov-exponent conjecture

Let the dynamics be a smooth enough ($C^{1+}$) diffeomorphism or vector field on a compact manifold. Say that wild historic points are absent when no point has wild historic behavior.

No-wild-historic-behavior conjecture. Absence of wild historic points implies that every invariant probability measure is either atomic or has only zero Lyapunov exponents. An analogous conclusion is conjectured for all smooth enough ($C^{1+}$) local diffeomorphisms away from a sufficiently regular (non-flat) critical or singular set.

The paper explains that Peixoto's theorem yields absence of wild historic points for an open and dense set of vector fields on compact orientable surfaces, so the vector-field assertion is meaningful mainly in dimensions at least three. The broader conclusion remains open.