Symbolic dynamics for non-uniformly hyperbolic flows

What the math gives to ML

The paper provides a transferable mechanism for representing non-uniformly hyperbolic flows, including flows with fixed points, as symbolic Markov dynamics with a continuous roof-time variable. Its distinctive technical asset is a velocity-weighted Hölder coding estimate: reconstruction errors become smaller near equilibria because they are multiplied by the local vector-field speed. A practical neural analogue is a hybrid symbolic-continuous world model or state-space model whose discrete transition code is Markovian and whose holding time is learned separately, with a state-dependent metric or reconstruction weight derived from the predicted flow speed. The strongest falsifiable prediction is that reconstruction error versus symbolic distance follows a power law with exponent \(\kappa\), and its prefactor scales linearly with \(\|f(x)\|\), rather than remaining uniform near fixed points.