2015/12/14 by Sergi Elizalde, Elizalde, Sergi, Katherine Moore +1
Computer Science · Mathematics · #05A05 #37M10 #Algorithms and Data Compression #Cellular Automata and Applications #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1512.04479
openalex publication_date 2015/12/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.