From Tangency to Fractals: Quadratic Dynamics in Nested Convex Geometry

Abstract

We study the dynamics generated by return maps associated with nested convex bodies and growing domains satisfying the geometric normal property in the plane. These maps are defined by transporting boundary points along normal directions to the surrounding domain and projecting them back onto the boundary of a subsequent convex set. We introduce a tangency condition between consecutive convex sets and show that it cancels the linear term in the local expansion of the transition operators. As a result, the dynamics near tangency points is governed by a quadratic normal form with an explicit coefficient depending on curvature and second order geometric data. This quadratic tangency law constitutes the central mechanism of the system. We prove that this nonlinear contraction leads to super exponential convergence toward the tangency set. In logarithmic coordinates, the dynamics becomes approximately affine, which allows for an interpretation in terms of iterated function systems (IFS) and explains the emergence of fractal limit sets. The theory is illustrated by several geometric configurations. Ford circles reveal a connection with continued fractions, nested ellipses yield Cantor-type limit sets, and configurations such as stadia and rounded triangles demonstrate the coexistence of linear and quadratic regimes. In purely quadratic settings with m independent branches, the limit set has similarity dimension in logarithmic coordinates, and an estimation of its Hausdorff dimension. From a broader perspective, the combination of symbolic branching and nonlinear contraction suggests potential connections with geometry, in particular in hybrid classical quantum information processing frameworks.