We propose to use Persistent Mutual Information (PMI), the Shannon information about the infinitely separated future stored in the past, to quantify strong emergent behaviour. We test this approach on data from the logistic map, and find that PMI captures the global periodicities of the system independent of whether the attractor is chaotic or not. In the area-preserving Standard Map PMI is found to grow indefinitely with resolution. We use information dimensions of the underlying spaces to interpret the scaling behaviour.
The scaling index is the information codimension (IC), a function of resolution and time separation. We find that for the fully-integrable case curves of IC at different resolutions collapse onto a single time separation plot. The well-defined rescaling for which this happens is related to the specific manner in which apparent causality is lost with time. We also investigate the effects of increasing non-linearity and interpret the results in terms of character of existing orbits.
Finally, we consider the Dobrushin distance between probability distributions as an alternative measure of emergence, and examine Toom's majority voter PCA.
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