INFINITE-DIMENSIONAL THURSTON THEORY AND TRANSCENDENTAL DYNAMICS I: INFINITE-LEGGED SPIDERS

The paper’s Theorem 1.1 states and proves the classification of exponential maps with escaping singular value at prescribed exponentially bounded address s (not (pre-)periodic) and potential t>t_s, via an infinite-dimensional Thurston–type fixed-point scheme with id-type maps and spiders; existence and uniqueness follow from strict contraction of the σ-map on a compact asymptotically conformal subset and an adjusted Banach fixed-point argument, yielding an entire map whose singular value escapes on rays with the required (s,t) data . The candidate (model) solution proves the same classification by constructing parameter rays G_s via the equation g_s^κ(t)=κ: local existence/uniqueness near infinity (contraction in logarithmic coordinates), global continuation using control near infinity (Squeezing-type arguments), and the converse classification for escaping parameters; this approach relies on the standard dynamic-ray theory and parameter-ray structure in the exponential family, consistent with the preliminaries summarized in the paper (dynamic rays and asymptotics) . Hence both are correct, with substantially different proofs (Thurston iteration vs. parameter rays).