SUB-ADDITIVITY OF MEASURE-THEORETIC ENTROPIES OF COMMUTING TRANSFORMATIONS ON BANACH SPACE

The paper proves sub-additivity h_µ(f∘g) ≤ h_µ(f)+h_µ(g) (Theorem A) by a nontrivial local-entropy/Bowen-ball covering argument in Section 5, culminating in the display that lets ε→0 to obtain the inequality (see the end of the proof of Theorem A) . It also establishes Theorem B: if Eu(x,f)=Eu(x,g) µ-a.e., then for small δ the local unstable manifolds coincide, and if either map has trivial center, additivity holds h_µ(f∘g)=h_µ(f)+h_µ(g) via a carefully constructed decreasing partition η subordinate to the common unstable foliation and conditional-entropy calculus . In contrast, the model’s Part 1 attempts to derive sub-additivity from a naive partition-join inequality that is not valid in general; indeed, the paper explicitly recalls that sub-additivity fails for commuting measure-preserving transformations without the differentiable/structural assumptions (Hu’s counterexample) . The model’s Parts 2 and 3 are directionally aligned with the paper (use of graph transforms/commutativity for plaque invariance, and a common unstable partition for additivity), but they omit critical hypotheses (finite box-counting dimension, quasi-compactness via l_α<0, and the integrability condition (H4)) and invoke a Ledrappier–Young u-entropy identity without verifying its applicability; the paper instead proves the needed partition equalities directly and uses conditional entropy identities to conclude additivity . The “no new positive exponents” intuition is made precise in the paper by a commuting MET and the identity for Lyapunov growth under f^s g^t (Eq. (3.1)) and the equality of unstable plaques is proved rigorously (Theorem 6.2 and the proof of Theorem B’s first part) . Overall, the paper’s arguments are coherent and complete under (H1)–(H4), while the model’s solution is incorrect in Part 1 and under-justified in Part 3.