LI-YORKE CHAOTIC WEIGHTED COMPOSITION OPERATORS

The paper’s Theorem 1 gives a necessary-and-sufficient criterion for Li–Yorke chaos of weighted composition operators C_{w,f} on C0(X) in terms of a sequence of relatively compact open sets (Bi) and conditions (A)–(B), and proves both directions using a residual-set argument and Banach–Steinhaus; it also extracts (A) from an irregular vector constructed via Bi := {x: |ψ(x)| > i^{-1}} and derives (B) by contradiction if sup_{i,n}‖w(n)‖_{f^{-n}(Bi)} were finite. This exactly matches the statement the model addresses, and the paper’s proof is sound (see Theorem 1 and its proof, including the Urysohn-lemma step and the residual/Banach–Steinhaus argument ). The model’s solution proves the same equivalence: the “only if” direction coincides with the paper’s reasoning, while the “if” direction constructs an explicit irregular vector using bump functions and a diagonal smallness subsequence, rather than appealing to Banach–Steinhaus. There is a minor technical nit: the model writes an equality (⋆) that should be an inequality (it does not affect the argument), and its convenience assumption of pairwise disjoint supports can be implemented by a mild refinement (or avoided entirely by following the paper’s residual approach). Overall, both are correct; the proofs differ in technique.