PERIODIC POINTS OF WEAKLY POST-CRITICALLY FINITE ALL THE WAY DOWN MAPS

The paper proves that for weakly PCF-all-the-way-down holomorphic endomorphisms, every eigenvalue along a periodic cycle is either 0 or has modulus >1, via an induction on dimension combined with (i) the unbranched/covering case on the universal cover and (ii) local germ lemmas at smooth invariant hypersurfaces and their intersections. The key steps are Theorem 3.5 (off the post-critical set and on the normal quotient at smooth points) and Theorem 3.6 (the full induction), supported by Propositions 2.1–2.2 and a reduction via proper linear projection in the singular-intersection case. The candidate solution mirrors this scheme point-by-point, including the same three intersection cases and the same lifting/normal-family argument off the post-critical set. See the paper’s abstract and setup of Theorem 1.1, Theorem 3.5 and its proof sketch steps, and the proof of Theorem 3.6 with cases and propositions, respectively .