Very interesting. Do you have any suggestions for reading more about this?
"It is actually one of the nice facts of life that as you dial up the local instability of a dynamical system you get a corresponding increase in the statistical stability."
I am a mathematician working in a tangentially related area and have never come across this before.
The most accessible introduction (in my opinion) is 'Laws of Chaos - Invariant Measures and Dynamical Systems in One Dimension' by Boyarsky and Gora. For a short survey on just stability of the invariant measure you could look at 'Linear Response, Or Else' by Baladi. I also like the lecture notes 'Statistical properties of uniformly hyperbolic maps and transfer operators' spectrum' by Carlangelo Liverani.
Does this perhaps have some relation to Monte Carlo methods where the qualities of the noise (white, pink blue etc) and/or stratification that you use - can strongly affect the qualities of the integral that results? Could be analogous to the local instabilities in the section you quoted?
(reminds me a lot of how information moves around in a reaction-diffusion simulation too)
It seems likely to. In MCMC this is known as the mixing/convergence speed. If one valid transition kernel decreases the mutual information between samples faster than another it mixes faster. Meaning the sampled chain has better statistical properties (effective number of independent samples).
"It is actually one of the nice facts of life that as you dial up the local instability of a dynamical system you get a corresponding increase in the statistical stability."
I am a mathematician working in a tangentially related area and have never come across this before.