Characterizations of extremal hyperbolic rates via Herglotz measures and Koenigs linearization

What the math gives to ML

The paper identifies a strong form of stable iterative dynamics: a disk self-map with boundary multiplier \(\alpha<1\) has exactly linear hyperbolic progress when its Koenigs coordinate remains non-degenerate at the attracting boundary point. The transferable asset is the conjugacy \(h\circ g=h/\alpha\), which turns nonlinear recurrent evolution into a controlled dilation and provides an explicit long-horizon growth rate \(\log(1/\alpha)\) rather than relying on unconstrained Jacobian tuning. A practical neural adaptation is a complex-valued recurrent or world-model latent state constrained to the unit disk, jointly learning a Koenigs coordinate and penalizing conjugacy error plus boundary degeneracy; this should improve rollout stability and make multi-step behavior predictable.