TOPOLOGICAL R-PRESSURE AND TOPOLOGICAL PRESSURE OF FREE SEMIGROUP ACTIONS

The paper establishes (A) lim_{r→0} Pr = P for free semigroup actions and (B) P(f0,…,fm−1,φ) = P(f0^{-1},…,f_{m−1}^{−1},φ) when the generators are homeomorphisms. Part (A) is proved via a skew-product reduction to the single-map case together with Chen’s result and the pressure identity PD(F,g)=log m+P (citations: Theorem 1.4 statement and proof sketch; Lemma 4.1; Theorem 2.1; Theorem 1.3) . Part (B) is obtained by showing PD(F^{-1},g)=PD(F,g) for the homeomorphism F and then using the same skew-product identity on the inverse semigroup (citations: Theorem 1.5 statement and its proof) . The candidate solution’s Part (A) provides a plausible and essentially correct direct covering argument using mistake balls and a combinatorial bound. However, Part (B) asserts an exact identity d_w(x,y)=d^{inv}_{θ(w)}(f_w x, f_w y) and consequent equality of weighted spanning quantities word-by-word; this fails in general for non-commuting generators, due to the suffix/prefix convention w′≤w used to define d_w and the nontrivial conjugations that arise in g_v f_w when v≤θ(w) (see the paper’s precise definitions of d_w and w′≤w) . Thus the model’s Part (B) proof is incorrect even though the statement itself is true by the paper’s skew-product argument.