Global Convergence Regime of the Return Dynamics in the Class O C

We study the global dynamics of the geometrically generated return map associated withdomains in the class OC . Starting from a convex core C ⊂ RN and a positive thicknessfunction d : ∂C → (0, ∞), the return construction defines a discrete self-mapF : ∂C −→ ∂Cthrough an outward normal displacement to the outer boundary followed by the inwardnormal first return to the core.Building on the local geometric theory of the return map, we derive a uniform globalone-step estimateF(c) − c = Bc∇∂C d(c) + O |∇∂C d(c)|2 Bc = d(c) I − d(c)Sc−1,under explicit smoothness, transversality, and small-slope assumptions. The tangential re mainder is shown to be of cubic order. This yields a second-order expansion of the thicknessincrement governed by the self-adjoint preconditioned HessianKc = B1/2c HcB1/2c.A directional positivity condition, and in particular the sufficient spectral boundλmin(Kc) ≥ −2 + δ,implies strict global thickness ascent,d(F(c)) − d(c) ≥ γ|∇∂C d(c)|2.Consequently,V(c) = dmax − d(c)is a strict Lyapunov function. We prove finite Lyapunov dissipation, square summabilityof the thickness gradient, vanishing return-step size, and, when the critical points of d areisolated, convergence of every orbit to a unique critical point. At nondegenerate attractingmaxima we obtain exponential convergence rates from the exact linearizationDF(c∗) = I + B∗H∗.Within the strict-ascent regime, nontrivial periodic and recurrent orbits are excluded.Finally, numerical computations performed with the exact unit-circle return formula showthat the spectral value −2 also marks the local loss of attraction. In a profile for whicha multiplier has crossed −1, the exact first-return branch remains globally real, negativethickness increments are detected, and a stable nontrivial period-two orbit is numericallyresolved. This provides a concrete exact-map example of the transition from curvature preconditioned global convergence toward periodic return dynamics