Existence of sequences with non-empty but measure-zero asymptotic layers

Determine whether there exist sequences on heteroclinic or homoclinic networks for which the refined asymptotic basin components D^(k)_δ(q) are non-empty sets of Lebesgue measure zero, and characterize such sequences if they exist.

Background

The paper introduces a decomposition of the asymptotic basin of attraction into layers Dk_δ(q), describing when a trajectory begins to follow a sequence. For preperiodic sequences, these layers eventually vanish entirely. The authors ask whether intermediate behaviors exist in which layers persist but have zero measure.

Answering this would refine the taxonomy of stability for sequences beyond finite and infinite types and reveal subtle measure-theoretic phenomena near networks.

References

We conclude by discussing some open questions, whose answers are beyond the scope of this note. Since for any preperiodic sequence q we not only have μ( D{(k)}_δ(q) ) = 0, but even D{(k)}_δ(q) = ∅ it is in this context also worth asking whether there exist sequences q such that D{(k)}_δ(q) is a non-empty set of measure zero.

— How many points converge to a heteroclinic network in an aperiodic way? (2410.11383 - Bick et al., 15 Oct 2024) in Section 6 (Discussion)