Permutations and β-shifts

Given a real number β > 1 , a permutation π of length n is realized by the β-shift if there is some x ∈ [ 0 , 1 ] such that the relative order of the sequence x , f ( x ) , … , f n − 1 ( x ) , where f ( x ) is the fractional part of βx, is the same as that of the entries of π. Widely studied from such diverse fields as number theory and automata theory, β-shifts are prototypical examples of one-dimensional chaotic dynamical systems. When β is an integer, permutations realized by shifts were studied in Elizalde (2009) [5]. In this paper we generalize some of the results to arbitrary β-shifts. We describe a method to compute, for any given permutation π, the smallest β such that π is realized by the β-shift. We also give a way to determine the length of the shortest forbidden (i.e., not realized) pattern of an arbitrary β-shift.