Synchronization and Applications of Piecewise Recursive Sequences with Dynamic Thresholds

The paper’s Common Limit Theorem (Theorem 5.7) asserts that, under continuity of f and g, infinite visitation of both regimes, and convergence of the three sequences {f(a_n)}_{n∈I1}, {g(a_n)}_{n∈I2}, and {c_n}, all three limits coincide. The system and visitation notions are set in Section 2 with Convention 2.1 and Definition 2.3 (and its equivalence to lim inf < 0 < lim sup), and a detailed statement appears in Lemma 5.6. The paper’s proof uses transition sets to deduce Lg ≤ Lc ≤ Lf (sound), but the key “Step 3” argument attempting to force Lf = Lg is incorrect: it relies on continuity in a way that does not preclude a continuous g from mapping points near Lf into a small neighborhood of Lg, and it further makes an unjustified inequality claim about Lc. A simple counterexample with f ≡ 1, g ≡ −1, h ≡ 0, a_0 = −1, c_0 = 0 yields infinite regime switching, convergence of all three sequences in the theorem’s statement, yet distinct limits: 1, −1, and 0. This directly falsifies Theorem 5.7 as stated. Because Corollary 5.8 and the introduction’s informal main theorem lean on this structure, the paper’s central claim is not established under its current hypotheses.