AN ANALOGUE OF GREEN’S FUNCTIONS FOR QUASIREGULAR MAPS

The paper proves existence of a nonnegative continuous function GK,θ,c with GK,θ,c∘H=2GK,θ,c, positive on the escaping set and zero on the bounded orbit set, by building a τ-semi-conjugacy via a Böttcher-type conjugacy near infinity and pulling back to all of I(H), then extending by 0 on BO(H) (Theorem 1.1; construction around Lemmas 3.5–3.9 and Theorem 2.5) . The candidate solution independently constructs G via the escape-rate limit G(z)=limn→∞2−n log+|Hn(z)|, justified using global growth bounds akin to Lemma 2.4 (|H(z)|≈|z|2 near infinity) to obtain uniform Cauchy convergence, the functional equation, and the correct zero/positivity sets . Both arrive at the same conclusion through different methods.