Permutations and beta-shifts

TL;DR

This paper explores the relationship between permutations and beta-shifts, providing methods to compute the minimal beta for realizing a permutation and identifying the shortest forbidden patterns in beta-shifts.

Contribution

It generalizes previous results from integer to arbitrary beta-shifts, introducing algorithms to determine minimal beta and shortest forbidden patterns for any permutation.

Findings

Method to compute the smallest beta for a given permutation

Algorithm to find the shortest forbidden pattern in a beta-shift

Extension of known results from integer to real beta-shifts

Abstract

Given a real number beta>1, a permutation pi of length n is realized by the beta-shift if there is some x in [0,1] such that the relative order of the sequence x,f(x),...,f^{n-1}(x), where f(x) is the factional part of beta*x, is the same as that of the entries of pi. Widely studied from such diverse fields as number theory and automata theory, beta-shifts are prototypical examples one-dimensional chaotic dynamical systems. When beta is an integer, permutations realized by shifts where studied in [SIAM J. Discrete Math. 23 (2009), 765-786]. In this paper we generalize some of the results to arbitrary beta-shifts. We describe a method to compute, for any given permutation pi, the smallest beta such that pi is realized by the beta-shift. We also give a way to determine the length of the shortest forbidden (i.e., not realized) pattern of an arbitrary beta-shift.