Permutations and shifts

The entropy of a symbolic dynamical system is usually defined in terms of the growth rate of the number of distinct allowed factors of length $n$. Bandt, Keller and Pompe showed that, for piecewise monotone interval maps, the entropy is also given by the number of permutations defined by consecutive elements in the trajectory of a point. This result is the starting point of several works of Elizalde where he investigates permutations in shift systems, notably in full shifts and in beta-shifts. The goal of this talk is to survey Elizalde's results. I will end by mentioning the case of negative beta-shifts, which has been simultaneously studied by Elizalde and Moore on the one hand, and by Steiner and myself on the other hand.