Rational Maps with Integer Multipliers

The uploaded note proves exactly the local rigidity statement the model labeled as likely open: Proposition 1 asserts that if g:U→V is an escaping quadratic-like map and the multipliers at all periodic points lie in OK (ring of integers of an imaginary quadratic field), then g is conjugate to an affine escaping quadratic-like map, and provides a complete proof via a differential-equation method and the discreteness of OK . The proof normalizes one branch to g1(z)=λ1 z, constructs a special sequence of periodic points z_n and multipliers ρ_n, derives an eventual exact linear relation ρ_n=a λ1^n + b using that OK is discrete, promotes this to a holomorphic differential equation g2'(z)=a + b g2(z)/z on the other branch, and then shows g2' is constant by an iteration/nonlinearity computation, hence g2 is affine . This local result, combined with Lemma 4, yields that the associated global rational map is exceptional, i.e., a power, Chebyshev, or Lattès map, re-proving Ji–Xie’s theorem without the non-archimedean input . The model’s claim that mere arithmetic membership in OK is too weak (and that one needs word-by-word multiplicative identities) is refuted by the paper’s argument, which extracts exact identities from ring discreteness and dynamics along a carefully chosen sequence.