On Dynamics of λ + tan z^2

The paper asserts that for λ in four unbounded quadrant regions Ai, fλ(z)=λ+tan(z^2) has an attracting fixed point capturing all singular values and the Julia set is a Cantor set topologically conjugate to a full two‑sided shift on a countable alphabet (Theorem 4.4 and Lemma 4.3). However, key steps are only sketched and contain errors: e.g., it incorrectly identifies the critical value as the origin early on (it is λ), conflicts with its later use of “critical value λ” ; it uses a fixed‑point theorem on a ball without rigorously showing fλ maps that ball strictly inside itself or that the fixed point is attracting; it claims complete invariance of the unbounded basin with a very rough path‑lifting argument; and it treats the two‑sided full shift over a countable alphabet as compact/Cantor, which is false (compactness/Cantor property requires a finite alphabet) . The model’s solution fills in some missing analytic estimates (asymptotics of tan on large imaginary parts) and provides a cleaner construction of a forward‑invariant disc that captures singular values, but it also contains gaps: it asserts a uniform Euclidean expansion on the complement using an incorrect inequality and relies on compactness of the two‑sided countable shift. Both arguments can likely be repaired by citing general expansion results for hyperbolic functions in the Speiser class and by formulating the conjugacy to the one‑sided full shift (or to the natural extension for the two‑sided shift), but as written both are incomplete.