Dimension of self-conformal measures associated to an exponentially separated holomorphic IFS

What the math gives to ML

The paper proves that, for an exponentially separated holomorphic iterated-function system without common fixed points, invariant analytic curves, or homothetic degeneracy, self-conformal measures attain the entropy-to-Lyapunov upper bound for dimension. The transferable mechanism is a quantitative relation between symbolic diversity and contraction: effective attractor dimension is bounded by branch entropy divided by average contraction rate, while overlaps reduce realized dimension. A practical transfer is to build a contractive multi-branch recurrent or generative network and regularize branch separation and entropy relative to contraction, testing whether its attractor dimension approaches the predicted value.