Patterns of Negative Shifts and Beta-Shifts

Abstract

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

Revised draft. 35 pages, 1 figure

References in corpus (1)

• The number of permutations realized by a shift

Cited by in corpus (1)

• Permutations and negative beta-shifts