Are all finite-type f.a.s. sequences preperiodic?

Establish whether every fragmentarily asymptotically stable sequence of finite type (f-f.a.s.) on a heteroclinic or homoclinic network is preperiodic.

Background

The authors define fragmentary asymptotic stability for sequences and refine it into finite type (f-f.a.s.) and infinite type (i-f.a.s.). They show that preperiodic sequences, if f.a.s., are always of finite type. A natural converse—whether all f-f.a.s. sequences must be preperiodic—would complete the classification and simplify stability checks to finite products of transition matrices.

Resolving this would align sequence stability with established criteria for cycles and omnicycles and clarify whether non-preperiodic (aperiodic) sequences can ever be finite-type f.a.s.