Statistics of Multipliers for Hyperbolic Rational Maps

The uploaded paper proves exactly the asymptotic claimed in the candidate solution. Theorem 2.1 states that for a hyperbolic rational map whose Julia set is not contained in a circle, and for sequences of intervals In and arcs Sn with sub-exponential inverse growth, one has π(n, α, In, Sn) ∼ [ν(Sn)/(σα√(2π))](∫In e^{−ξα x} dx) e^{H(α)n}/n^{3/2}, and the shrinking-interval corollary with midpoint pn, matching the candidate’s displays . The paper’s proof employs Markov partitions and the distortion/rotation observables r(z)=log|f′(z)| and θ(z)=arg f′(z), together with twisted Ruelle operators L(s,k)=L_{sR+ikθ}, where R=r−α . It establishes Dolgopyat-type decay for all twisted modes (b,k)≠(0,0), leveraging Oh–Winter’s non-local-integrability to obtain a spectral gap and estimates for Zn(ξ+ib,k) that suppress nonzero Fourier modes in the S^1 variable . The main term arises from the k=0, small-b regime via a Taylor expansion e^{P((ξ+it)R)}=e^{P(ξR)}(1−σ^2_α t^2/2+O(|t|^3)) and a dominated-convergence/Gaussian integral, producing the 1/√n factor with variance σ^2_α>0, exactly as in the model’s saddle-point argument . The passage from fixed points to primitive orbits is handled with a 1/n factor and an exponentially small error, as the candidate also states . Thus the candidate reproduces the paper’s approach: smoothing of indicators and Fourier–Laplace representations (eqs. (4.2)–(4.3)) , uniform twisted spectral bounds, and a local limit analysis near η=0. The definitions of H(α), ξ_α, and σ_α with σ_α^2>0 under the non-circle hypothesis also align with the paper’s setup .