arXiv:2608.19393 2026 Dynamics 2 ideas extracted · analyzed Sep 1, 2026
The paper provides a constructive class of contact billiard maps whose dynamics on invariant quadratic leaves reduces exactly to iterating a fixed low-dimensional linear transformation. Depending on the quadratic table and leaf parameter, the transformation is elliptic, hyperbolic, or loxodromic, with explicit rotation angles, reciprocal expansion rates, and invariant quadratic structures. This is transferable to recurrent and state-space neural architectures as an analytically controlled memory layer: preserve selected invariants, use reciprocal gains for tunable long memory, and avoid unconstrained recurrent matrices. The main falsifiable predictions are exact norm preservation in the elliptic regime and predictable exponential growth and decay rates in the loxodromic regime.
Use paired recurrent channels with exactly reciprocal gains while applying a common phase rotation. One channel carries a controlled expanding mode and the other a matching contracting mode, creating a tunable hyperbolic memory spectrum without the optimization fragility of an unconstrained recurrent matrix.
Replace an unconstrained recurrent transition with block-diagonal planar rotations whose angles are learned or conditioned on a slowly varying context variable. The resulting hidden-state norm and each two-dimensional block energy are exactly invariant in the ideal recurrence, preventing exploding or vanishing recurrent dynamics while retaining phase information over long horizons.