Matching of observations of dynamical systems, with applications to sequence matching

The paper defines Yi = −log max_{j=2..q} d(f(T^i x1), f(T^i xj)) on the product system, chooses thresholds via the generalized dimension D^f_q so that μ^q(Y0>u_n(s)) ~ e^{-s}/n, and identifies Fn(u_n) with a survival/hitting-time probability for the shrinking targets Sq_n, then applies the Keller–Liverani spectral rare-event theory to obtain Fn(u_n(s)) → exp(−θ^f_q e^{−s}) (Proposition 1), with θ^f_q given by short-return probabilities pk,q (eqs. (10)–(12)). This exactly matches the model’s argument: same threshold normalization using D^f_q, the same hitting-time equivalence, and the same spectral small-hole expansion and pk,q representation of the extremal index. Minor differences are only in presentation and assumptions (the paper phrases conditions via REPFO or mixing schemes Д1/Д′1, while the model assumes a spectral gap for the product operator), but the proofs are substantively the same. See the paper’s setup and Proposition 1, including the choice of u_n(s) via Definition 2 of D^f_q and the extremal index representation (, , , ).