Thermodynamic formalism for intermittent maps with multiple neutral fixed points and phase transitions

What the math gives to ML

The paper provides a constructive mechanism for systems with several intermittently neutral fixed points: trajectories can spend power-law residence times near different fixed points, while a pressure variational principle selects between entropy-rich and low-entropy phases. Near a neutral point, the local displacement scales as |f(x)-x| approximately b_k|x-xi_k|^(1+alpha_k), producing polynomial rather than exponential escape. A transferable neural-network construction is an intermittent multi-mode residual or routing state whose neutral modes preserve information for long periods, with a temperature-like parameter controlling an entropy-versus-preference phase transition. The key falsifiable signatures are power-law residence-time tails and a sharp change in mode occupancy or entropy when competing pressure branches cross.