A surprising new application of pattern-avoiding permutations to dynamical systems is that they can be used to distinguish random from deterministic time series.

The orbits generated by piecewise monotone maps on one-dimensional intervals always have forbidden patterns, that is, ordered subsequences that do not occur in any orbit. Besides, if a pattern is forbidden for a given map, then any longer permutation that contains it as a consecutive pattern is forbidden as well. On the other hand, in a random time series, every pattern appears with some positive probability, which approaches one as the length of the time series increases. This idea can be used to create tests to distinguish random from pseudo-random dynamics.

The second part of the talk will focus on shift maps, where one can apply combinatorial tools to describe and enumerate their forbidden patterns.