Entropy and semiconjugacy on surfaces

What the math gives to ML

The paper provides a concrete semiconjugacy mechanism: when a smooth surface map has the same topological entropy as a pseudo-Anosov reference map in the same isotopy class, its dynamics factor through the reference map via a continuous map satisfying \(\pi\circ g=f\circ\pi\). The nontrivial structural result is that every fiber \(\pi^{-1}(x)\) is connected, represented as a nested intersection of closed topological disks, and that the maximal-entropy measure of the reference system has a unique invariant lift which is metrically isomorphic to it. A transferable neural-network version is a latent-dynamics architecture that separates entropy-carrying macroscopic dynamics from connected, dynamically redundant fibers.