Topological synchronisation or a simple attractor?

The paper proves that for Collet–Eckmann S‑unimodal maps with a quadratic critical point and slow recurrence, the acip’s generalized dimensions satisfy D_q = 1 for q < 2 and D_q = q/(2(q−1)) for q ≥ 2, relying on a density decomposition with square‑root cusps and a Legendre‑transform characterization of D_q; see Proposition 5.1 and the density form h(x)=ψ0(x)+∑k≥1 φk(x)/√|x−z_k| with exponentially decaying amplitudes. The candidate model derives the same spectrum by directly estimating I_q(r)=∫μ(B(x,r))^{q−1} dμ(x), splitting contributions from regular regions and cusp neighborhoods. The conclusions match, with the model offering a more direct integral computation and the paper using a multifractal-formalism route. Minor differences are present (e.g., the model should weight cusp neighborhoods by decaying amplitudes), but they do not affect the final result. Overall, both are correct and essentially consistent in assumptions and outcome.