Decay of correlations for the inverse-limit extension without a bounded density

Establish whether the inverse-limit dynamical system $(X_\infty,T_\infty,\mu_\infty)$ associated with the tower of finite sheeted covers has decay of correlations when the absolutely continuous invariant probability density $h$ of the base system is not bounded away from zero.

Background

Theorem 1 proves uniform exponential decay of correlations along finite covers under suitable expansion, monodromy, and transfer-operator assumptions. The paper then passes to the inverse limit and notes that the finite-level estimates imply exponential decay for the inverse-limit system when the base density is bounded away from zero.

When the density is not bounded away from zero, the authors can still prove persistence of variance properties, including existence, continuity, and coboundary characterization. They explicitly leave open whether the corresponding decay-of-correlations statement remains valid.

References

Without the assumption that the ACIP density $h$ is bounded away from $0$, we don't know whether such a decay of correlations result holds for $T_\infty$, but what we do find is that a good understanding of variance persists.

— Relative ($τ$), Expanders, and Decay of Correlations for certain Expanding Maps  (2609.05271 - Dougall, 4 Sep 2026) in Section 1, subsection “Central limit theorem and variance for the inverse limit”