SATURATED PROPERTIES AND OPTIMAL ORBITS FOR CHAOTIC SYSTEMS

The paper’s Theorem 1.1 and 1.2 state exactly the equalities the model aims to prove, for upper capacity (h^{UC}_{top}) and packing (h^{P}_{top}) entropies of (transitively) saturated sets under g–almost product and uniform separation properties . The proofs build closed sets via a nested gluing/intersection scheme and a δ*–ε* uniform-separation estimate to guarantee both separation and that the constructed points truly lie in (transitively) saturated sets, e.g., the G(N)_s construction and its separation argument in Section 3 , with preliminaries on g-APP and entropies in Section 2 . By contrast, the model’s lower-bound constructions glue a finite prefix in front of a fixed “skeleton” x*∈G_K and then assert y(v)∈G_K by invoking “finite prefixes do not change M_x.” This step is invalid without ensuring T^m y(v)=x* (not provided by g-APP nor by any surjectivity assumption). The paper avoids this gap by using a diagonal/nested concatenation scheme ensuring membership in G_K (and in GT_K when needed) . For the packing upper bound, the model also appeals to uniform separation to upper-bound the cardinality of disjoint Bowen balls, but the paper instead uses general upper bounds from Feng–Huang’s variational framework and local measure-theoretic upper entropy to control packing sums (cf. Lemma 2.2 and the Section 4 argument) . Minor editorially, the paper’s statements include “>0” in Theorem 1.1; the equality holds regardless of the sign of h_{top}(T,X), and “>0” is only automatic when h_{top}(T,X)>0 (a nuance the model notes).