R- and C-supercyclicity for some classes of operators

The paper proves a shift-like characterization: for dissipative composition operators with bounded distortion, Tf is C-supercyclic if and only if for any fixed q, lim_{n→∞} min{μ(f^{q−n}(W)), μ(f^{q+n}(W))} = 0, obtained by factoring Tf onto a bilateral weighted backward shift with weights w_k = (μ(f^{k−1}(W))/μ(f^{k}(W)))^{1/p} and invoking the standard supercyclicity criterion for weighted shifts . The candidate asserts a strictly stronger and generally false equivalence, namely that Tf is C-supercyclic iff μ(f^{q−n}(W))·μ(f^{q+n}(W)) → 0 for every q. A simple counterexample already contained in the paper’s framework shows the flaw: take a system whose associated shift has constant weight 0 < w < 1 (so ν(i)=μ(f^i(W))/μ(W) grows exponentially in one direction and decays in the other). Then min{μ(f^{q−n}(W)), μ(f^{q+n}(W))} → 0, so Tf is C-supercyclic by Theorem 4.8, but μ(f^{q−n}(W))·μ(f^{q+n}(W)) stays bounded away from 0 (indeed constant), falsifying the candidate’s “iff” claim . The paper’s route via factorization (Lemma 4.1), shift–composition conjugacy (Proposition 4.3), and the classical shift criterion (Theorem 4.7) is standard and consistent, whereas the model’s derivation of a product-limit condition misidentifies the necessary-and-sufficient asymptotic.