Infinite-Piecewise Expanding Maps: Chaos, Ergodicity and Invariant-Set Complexity

What the math gives to ML

The paper provides a constructive countable-branch iterated-function-system mechanism: uniformly contracting inverse branches encode trajectories by sequences over a countably infinite alphabet, while the induced expanding map is conjugate on its invariant set to the full shift on \(\mathbb{N}^{\mathbb{N}}\). This offers a transferable design for neural modules with adaptive, potentially unbounded routing or hierarchical latent codes, where each selected branch contracts representation uncertainty but the branch sequence preserves combinatorial memory. The key engineering certificate is explicit: an \(n\)-step branch composition has sensitivity at most \(s^n\), with \(s<1\), while branch derivative regularity controls distortion. The most practical first use is a finite-top-\(K\) approximation of a countable mixture-of-experts or recurrent latent-state module with contraction enforced per branch.