SRB Measures for $C^{1+\mathrm{Dini}}$ Diffeomorphisms
arXiv:2607.13530
2026
Regularization
1 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The paper provides a quantitative bridge between the geometry of unstable manifolds and entropy production: for an invariant measure, entropy is bounded by the sum of positive Lyapunov exponents, and equality characterizes SRB measures under a dominated splitting and C1+Dini regularity. The transferable asset is an entropy-Lyapunov gap that can be monitored or regularized in recurrent neural networks and learned dynamical models. A practical adaptation is to estimate finite-time entropy rate and Jacobian Lyapunov exponents during training, penalizing unexplained entropy deficit only when the model is intended to represent a natural chaotic invariant measure.
Ideas from this paper
Unverified
2026
Add an entropy-Lyapunov consistency term to a recurrent or state-space model whose learned dynamics are intended to reproduce a chaotic invariant distribution. The regularizer targets the equality condition h_mu(f) = sum_i max(lambda_i, 0), while a dominated-splitting diagnostic determines whether the theorem assumptions are approximately plausible instead of blindly forcing equality.
Useful6/10
Difficulty6/10
Novelty8/10