Degenerate fixed points of maps in Banach spaces and lattices with decay and their invariant manifolds

What the math gives to ML

The paper offers a constructive parametrization method for invariant manifolds near degenerate fixed points of maps on Banach spaces, including infinite lattice systems with decay. Its transferable mechanism is the invariance equation F composed with K equals K composed with R, which represents a high-dimensional dynamical system through a lower-dimensional embedded manifold and a reduced map, even when linearization is nonhyperbolic. A practical neural transfer is to constrain recurrent or state-space hidden dynamics to approximately preserve a learned slow manifold, while weighted norms enforce decay across spatial sites or memory channels. The main falsifiable signature is geometric: transverse deviations should decay at a rate determined by the normal spectral radius, while latent errors follow the slower reduced dynamics.