Equivalence between ACIM instability and slow-time-scale Markov behavior

Establish whether weak convergence of the invariant measures of a family of piecewise C2 expanding maps to a nontrivial mixture of an absolutely continuous measure and a singular measure is equivalent to the emergence, on an appropriate slow time scale, of an asymptotic jump Markov process whose localized states are finite critical orbits.

Background

The paper proves that, for the class of families studied in the main body, metastable motion between an extended state and a shrinking localized state converges on a slow time scale to a jump Markov process. In the multiple-turning-point appendix, the localized states correspond to neighborhoods of critical fixed points and the limiting singular measures are concentrated on those points.

The second question asks whether this relationship is a general equivalence rather than a feature of the paper's framework. It concerns families satisfying the global regularity assumptions (C0) and (C5), with the limiting invariant density positive at at least one periodic critical point, and allows the singular component to be a convex combination of measures supported on finite critical orbits.