ON INVARIANT MEASURES OF "SATELLITE" INFINITELY RENORMALIZABLE QUADRATIC POLYNOMIALS

The paper’s main theorem explicitly proves that if f(z)=z^2+c has infinitely many satellite renormalizations, then the restriction f: J_∞ → J_∞ carries no invariant probability measure with positive Lyapunov exponent, and in fact any ergodic invariant measure supported on J_∞ must be supported on the postcritical set P with zero exponent (Corollary 1.1), see Theorem 1.1 and its consequences in the introduction and Section 6 . A key fact used in the paper is that all periodic points of f are repelling (hence |(f^{p})'(z)|>1 at such points), not parabolic, for the dynamical maps under consideration (item (B) in Section 2) . By contrast, the model solution critically assumes that each satellite renormalization F_n = f^{p_n} has a parabolic fixed cycle W_n with |F_n'(w)|=1 and that deeper small Julia sets eventually lie in any neighborhood of W_n; this is neither stated nor true in the paper’s setting, and it contradicts the repelling nature of the α/β fixed points for f^{p_n}: f(J_n)→f(J_n) described in the preliminaries . The model’s bound |log|F_n'(z)||≤ε on J_∞ near W_n therefore fails, collapsing the integral estimate. The paper’s proof instead uses a refined external-ray/sector construction (e.g., Lemma 2.2 on shrinking angular windows in the non-doubling case) and univalent pullbacks to contradict the existence of a positive-exponent invariant measure on J_∞ .