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Finite prefixes of aperiodic f.a.s. sequences

Determine, for a given aperiodic fragmentarily asymptotically stable sequence q on a network, which finite sequences can be prepended to q so that the resulting concatenated sequence remains fragmentarily asymptotically stable.

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Background

Since basins of attraction for sequences are forward-invariant only up to shifts, understanding how finite modifications at the beginning of a realized itinerary affect stability is nontrivial. This question probes the sensitivity of f.a.s. aperiodic sequences to finite prefixes.

Answering it would clarify the robustness of aperiodic f.a.s. sequences and help classify which finite concatenations preserve positive-measure attraction.

References

We conclude by discussing some open questions, whose answers are beyond the scope of this note. More generally, given an aperiodic f.a.s. sequence q, which (finite) sequences can we put in front of q such that the resulting sequence is also f.a.s.?

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