An Exactly Soluble Three-Dimensional Toy Model with a Saddle Separatrix
Year: 2026
Type: article
Abstract: We construct an explicit holomorphic function Gt(s)G_t(s)Gt(s) extending the real isosceles reduction of the pedal map associated with the Jonah triangle. At the parameter value t=7/10t=7/10t=7/10, we prove that the Jonah point s∗=2/7s_* = 2/7s∗=2/7 is a fixed point of GtG_tGt with multiplier λ=451/343\lambda = 451/343λ=451/343. Since ∣λ∣>1|\lambda| > 1∣λ∣>1, Koenigs' theorem implies that the dynamics is locally holomorphically conjugate to a linear dilation near s∗s_*s∗. This result establishes that the local complex dynamics around the Jonah triangle is trivial and non-fractal, shifting the focus of potential fractal behavior to the global structure of the map. The paper provides exact algebraic derivations of the germ and its derivative, distinguishing clearly between local analytic properties and global dynamical conjectures. No claim is made regarding the extension of GtG_tGt to the entire Riemann sphere CP1\mathbb{CP}^1CP1.
Author: Schloss Jonah
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225597323