Affine Anosov Maps on $\mathbb{R}^n$: Classification, Index Spectrum, and Stability at Infinity

What the math gives to ML

The paper provides a noncompact hyperbolicity mechanism: an affine map can be Anosov in a complete, state-dependent Riemannian metric even when its linear part has eigenvalues on the unit circle, provided the translation has a nonzero component in the neutral direction. The key condition is that the affine map has no fixed point, equivalently v is not in the image of I minus A, while the fixed-point case requires A itself to be hyperbolic. A transferable neural-network construction is a drift-assisted recurrent or state-space layer whose hidden-state metric is explicitly modulated by a monotone clock coordinate. This could stabilize long-horizon dynamics without forcing every Euclidean Jacobian eigenvalue strictly inside the unit disk.