Spectral Stability, Fatou Basin Contraction, and Machine-Checked Multipliers in Holomorphic Extensions of Generalized Syracuse Dynamics
Volkan Dağlı, Zerrin Dağlı, Dağhan Dağlı, Emrah Yurtseven, Murat Emre Özkaya
原文摘要
We embed the Syracuse operator $T(n) = n/2$ (even $n$), $T(n) = (3n+1)/2$ (odd $n$) and the Collatz map $C(n)$ into $\mathbb{C}$ via entire functions of exponential type $π$: $f(z) = \frac{1}{4}(1 + 4z - (1 + 2z)\cos(πz))$ and $F(z) = \frac{1}{4}(2 + 7z - (2 + 5z)\cos(πz))$. We prove five results: (1) every integer periodic cycle $C \subset \mathbb{N}^+$ of $T$ is a strictly attracting Fatou cycle of $f$ with multiplier $μ_f(C) = 3^{N_{\mathrm{odd}}}/2^N = \prod_{n \in C\text{ odd}} \frac{3n}{3n+1} \in (0,1)$, and $μ_f(O_{1,2}) = μ_F(O_{1,4,2}) = 3/4 < 1$; (2) on the Fatou basin $B(O_{1,2})$, $g = f^{\circ 2}$ conjugates to $Φ_j(g(z)) = \frac{3}{4}Φ_j(z)$ with Koenigs Taylor jets $c_2(1) = -\frac{13}{2}π^2$ and $d_2(2) = -\frac{9}{2}π^2$; (3) the Schwarzian derivative is strictly negative on $\mathbb{Z}$ ($S(f)(n) \le 3π^2 - \frac{3}{8}π^4 \approx -6.9196 < 0$), and a repelling Julia 2-cycle $O_{\mathrm{rep}} \subset J(f)$ ($μ\approx 1.2148 > 1$) isolates $B(O_{1,2})$; (4) on $(\mathbb{Z}_2, μ_{\mathrm{Haar}})$, the 2-adic Lyapunov exponent $λ_{\mathrm{Haar}} = \frac{1}{2}\ln(3/4) \approx -0.143841 < 0$ equals $\frac{1}{2}\ln|μ_f(O_{1,2})|$; and (5) we certify 20 theorems in Lean 4 (anc/CollatzAttractor.lean, 0 sorry). These results establish local Fatou invariants and 2-adic ergodic averages for $f(z)$, leaving open whether every $n_0 \in \mathbb{N}^+$ lies in $\bigcup_{k \ge 0} f^{-k}(B(O_{1,2}))$.
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