Invariant Measures in Time-Delay Coordinates for Unique Dynamical System Identification

The paper proves two main theorems: (i) equality of a single (m+1)-delay invariant measure implies a topological conjugacy on the supports (Theorem 3.1), and (ii) with a shared invariant measure µ, equality of (m+1)-delay invariant measures for m observables plus a short matching-orbit condition forces T=S on supp(µ) (Theorem 3.2). The statements and their proofs—via prevalence-based delay-embedding, identification of the (m+1)-delay image as the graph over the first m coordinates, and a conjugating homeomorphism Θy—are explicit in the paper (Theorem 3.1 statement and proof: , , construction of Θy and the key equal-support step using Lemma 3.1: , ; time-delay definitions: ; prevalence and embedding background: , ). The candidate solution follows the same backbone: uses Sauer–Yorke–Casdagli prevalence and periodic-point nondegeneracy to get injectivity of Ψ^(m), notes that Ψ^(m+1) is a graph defining a predictor, identifies equal supports from equality of pushforwards, and builds the conjugacy. For uniqueness (Theorem 3.2), the model also uses a prevalent choice of Y so each scalar delay map is injective and the static map y=(y1,…,ym) is injective, then extends orbit equality from the short matching window to all times and concludes T=S on supp(µ). These steps match the paper’s Lemmas and proof structure (Theorem 3.2 statement and proof sketch: ; use of Θy for each yi and evaluation at x*: ; final equality on the support via Lemma 3.3: ). The model adds a cosmetic predictor/left-shift viewpoint and a redundant note that x* is µ-typical for S, but these do not conflict with the paper. Hence both are correct and substantially the same.