Pseudo-Böttcher Components of the Wandering Set of Inner Mappings

The paper proves a sharp topological classification (Theorem 5.0.13) of totally invariant Pseudo‑Böttcher components S: S is planar and its ideal boundary splits into AIB and RIB, both nonempty; if Sing(f)∩S=∅ then each side is a singleton; if |Sing(f)∩S|=1 then AIB is a singleton and RIB is a Cantor set; otherwise RIB is Cantor and AIB is either a singleton or a countable set of isolated points. The proof uses the τ-structure, atoms/molecules/chains, the Conrod–Reeb graph (a tree), and Riemann–Hurwitz, and it employs canonical representations of ends via components of ωn (superlevel sets) . The candidate solution reaches the same classification using a different framework: a component graph of Wk:=τ−1([k,∞)), inverse‑limit paths as ends, and a branching argument to obtain Cantor structure. Substantively, the model’s classification agrees with the paper. There are two issues to note. (i) The model states globally that each level set τ−1(t) is a disjoint union of circles; in the presence of singularities, the paper shows certain level sets can be singular (a wedge of circles), so the model’s statement is too strong but fixable by restricting to regular levels or to atoms without singular points . (ii) For the dynamics on ends, the paper focuses on topology (Cantor/isolated) and only implicitly uses the induced map on RIB to prove closure/no isolated points via canonical representations ; the model goes further, asserting that main ends are fixed and auxiliary ends are preperiodic to main ends, which follows from the paper’s chain machinery (auxiliary forward chains become main under some iterate), but the model’s brief argument includes a nonessential misstep (treating f(Wi)=Wi−1 for components). Correcting that, the fixed/preperiodic claims can be justified via cofinal canonical representations and the paper’s “auxiliary-to-main under iterates” fact. Overall, the paper’s classification is correct and complete; the model’s solution is essentially correct (with a small technical correction) and uses a different proof style.