Properties of resonant states for generic smooth expanding maps

What the math gives to ML

The paper develops a genericity mechanism for resonant states of smooth expanding maps: resonances are generically simple, real resonant states are Morse functions, and zero is a regular value. The most transferable construction is a finite-dimensional observability criterion: evaluations of distinct real and imaginary modes at sufficiently many points form an invertible matrix. A neural-network analogue can regularize learned recurrent or state-space dynamics so that estimated Koopman modes remain linearly independent and geometrically nondegenerate. The method is falsifiable through singular-value gaps, eigenvalue separation, critical-point Hessians, and a predicted rank transition when the number of informative observation points reaches the number of modes.