Patterns of Negative Shifts and Beta-Shifts

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.

No reviews yet.

No comments yet.