Abstract: The β-shift is the transformation from the unit interval to itself that maps x to the fractional part of βx. Permutations realized by the relative order of the elements in the orbits of these maps have been studied for positive integer values of β and for real values β>1. In both cases, a combinatorial description of the smallest positive value of β needed to realize a permutation is provided. In this paper we extend these results to the case of negative β, both in the integer and in the real case. Negative β-shifts are related to digital expansions with negative real bases, studied by Ito and Sadahiro, and Liao and Steiner.
Follow this link to the arXiv preprint.