Cohomological Reduction for Fiber-Contracting Extensions:From Subcohomology to Thermodynamic Formalism

What the math gives to ML

The paper provides a constructive cohomological reduction for dynamical systems with uniformly contracting fibers: a state-dependent potential can be written as a quotient potential plus a telescoping coboundary, \(\varphi=\psi\circ\pi+u-u\circ F\). This means long-run invariant averages, minimizing measures, pressure, and equilibrium states are completely determined by the noncontracting quotient, while the stable fiber contributes only a finite-horizon boundary term. The most transferable neural-network mechanism is to separate a recurrent model into a slow/base state and a provably contracting auxiliary state, then train or optimize the quotient objective instead of the full state-dependent objective. The affine skew-product example gives an immediately implementable construction and predicts geometric decay of the discrepancy at rate \(|a|^t\) and finite-horizon objective errors of order \(1/T\).