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Stability criteria via infinite products of transition matrices or iterated function systems

Develop explicit conditions for fragmentary asymptotic stability of (possibly aperiodic) sequences in terms of infinite products of transition matrices or the associated iterated function systems, generalizing finite-product criteria known for preperiodic sequences.

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Background

For preperiodic sequences, stability can be analyzed using finite products of transition matrices derived from local maps near equilibria. Extending such criteria to aperiodic sequences would likely require handling infinite products or associated iterated function systems.

The authors pose this question to bridge classical cycle stability analysis with the new sequence-based framework, aiming for computable criteria for aperiodic itineraries.

References

We conclude by discussing some open questions, whose answers are beyond the scope of this note. More generally, is it possible to obtain similar conditions for some infinite product of transition matrices and/or the iterated function system they represent?

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