Shadowing Endomorphisms of Compact Groups

What the math gives to ML

The paper provides a sharp shadowing criterion for compact-group endomorphisms: after reduction to the surjective stable image, every finite-dimensional invariant sector of the rational dual dynamics must be hyperbolic, and the induced map on simple factors must have no periodic point. The transferable mechanism is the exclusion of neutral unit-modulus dynamics, which otherwise allows small local errors to accumulate into large trajectory deviations. This can be implemented in recurrent or state-space networks by monitoring Jacobian spectra, enforcing a spectral gap around the unit circle, and directly testing pseudo-orbit shadowing under bounded transition perturbations.