DYNAMICS OF THE MEROMORPHIC FAMILIES fλ = λ tanp zq

The paper proves the hyperbolic dichotomy for f_λ(z)=λ tan^p(z^q): if the immediate basin B0 contains the asymptotic value(s) then B0 is completely invariant, infinitely connected, and the Julia set is a Cantor set with shift dynamics; otherwise all Fatou components are simply connected and, for hyperbolic maps, J is connected and locally connected (Theorems A–C). The candidate solution establishes the same dichotomy via a slightly different route (Riemann–Hurwitz on the double and a countable IFS). One minor flaw is the claim that passing to f^2 leaves only one finite asymptotic value in the p odd case; this is unnecessary and incorrect, but the conclusion still follows using the paper’s symmetry lemma and capture-of-singular-values facts. Overall, the results and structural conclusions match the paper’s theorems and proofs.