Characterization of ACIM instability by local expansion near critical orbits

Determine whether a family of piecewise expanding maps satisfying conditions (C0) and (C5), whose limit map has exactly one finite orbit containing a critical point, is ACIM unstable only if there exists an eventually periodic point b_epsilon converging to that orbit and the local expansion along the critical orbit is at most one.

Background

The paper studies instability of absolutely continuous invariant measures for families of piecewise expanding interval maps. Its main results analyze a family with a critical turning fixed point and show that the limiting invariant measure may be absolutely continuous, singular, or a convex combination of an ACIM and a Dirac mass, depending on a local expansion quantity.

The first question seeks to extend this analysis from a single critical turning fixed point to a general finite orbit containing a critical point. It asks whether instability requires an eventually periodic point of the perturbed maps to approach that orbit and whether the product of the local expansions along the orbit must be at most one. The question is presented as a refinement of Keller's original conjecture about the mechanism responsible for ACIM instability.

References

Is it true that the family is ACIM unstable only if there is an eventually periodic point $b_\varepsilon$ such that $\lim_{\varepsilon\to 0} dist(b_\varepsilon, \mathcal{O})=0$ and the local expansion $Exp(\mathcal{O})$ of $T_0$ along $\mathcal{O}$ is at most $1$?

— ACIM instability of piecewise expanding maps through the lens of metastability  (2609.11704 - Komálovics et al., 10 Sep 2026) in Section Conclusions and questions, first Question environment