Equilibrium Stability for Open Zooming Systems

The paper states and proves Theorem A: FZ = {(f, φ) : f is zooming and φ has locally Hölder induced potential} is equilibrium stable, using a direct variational approach. The proof extracts an invariant weak-* limit, uses a generating partition for the limiting equilibrium (Lemma 3.0.1), upper semicontinuity of entropy (via Brin–Katok/local entropy and generating partitions), and a semicontinuity bound Pf(φ) ≤ lim sup P_{fn}(φn) to conclude that every weak-* limit μ0 is an equilibrium state for (f, φ) . In contrast, the model’s argument assumes one can choose, for all large n, a common inducing scheme with uniform expansion/distortion and a common coding that yields a uniform Lasota–Yorke inequality and spectral gap, then applies tower-level spectral stability and projects back. Those uniformity and common-coding assumptions are not guaranteed by the definition of FZ and are not used (nor needed) in the paper’s proof. Hence, the model solution relies on unproven hypotheses in this generality, while the paper’s argument is correct under the stated setting .