AN INVERSE PROBLEM IN PÓLYA–SCHUR THEORY. II. EXACTLY SOLVABLE OPERATORS AND COMPLEX DYNAMICS

The paper’s Proposition 6.4 proves that, for T(x−z)=zU(x)−V(x) with gcd(U,V)=1 and a non-exceptional rational map R=V/U of degree ≥2, the minimal closed T1-invariant set MT1 equals the plane Julia set JC(R). The proof shows (i) JC(R)∈IT1 via complete invariance (Lemma 6.2), and (ii) any closed S∈IT1 contains JC(R) by propagating backward orbits inside S and using that J(R) lies in the closure of a backward orbit . The candidate solution mirrors the key translation S∈IT1 ⇔ R−1(S)∩C⊆S and JC(R)∈IT1, but establishes JC(R)⊆S by a different route: it proves O:=Ĉ\S is forward-invariant, notes S must be infinite (non-exceptional ⇒ no finite exceptional points in C), selects three points a,b,c∈S, and invokes Montel’s three-value normality criterion to deduce O⊆F(R), hence JC(R)⊆S. This is a distinct, standard complex-dynamical argument, rather than the paper’s backward-orbit closure argument .