Statistical properties for irregular observables in slowly mixing hyperbolic systems

What the math gives to ML

The paper provides a constructive blocking and cylinder-set approximation mechanism showing that highly irregular indicator observables can still satisfy strong statistical limit laws in slowly, polynomially mixing hyperbolic systems. Its transferable asset is not the billiard geometry itself, but a quantitative recipe for separating temporal blocks, replacing discontinuous events by finite-resolution measurable approximations, and controlling the almost-sure approximation error. A promising neural-network use is mixing-aware training and evaluation for recurrent, state-space, or world-model systems whose losses and diagnostics contain threshold events, where naive iid minibatch assumptions underestimate gradient variance and long-horizon uncertainty.