Beta-Expansions with Negative Bases

Abstract

This paper investigates representations of real numbers with an arbitrary negative base ˇ< 1, which we call the . ˇ/-expansions. They arise from the orbits of the . ˇ/-transformation which is a natural modification of theˇ-transformation. We show some fundamental properties of . ˇ/-expansions, each of which corresponds to a well-known fact of ordinaryˇ-expansions. In particular, we characterize the admissible sequences of . ˇ/-expansions, give a necessary and sufficient condition for the . ˇ/-shift to be sofic, and explicitly determine the invariant measure of the . ˇ/-transformations.

SUBSTITUTIONS OVER INFINITE ALPHABET GENERATING (−β)-INTEGERS

“…The (−β)-expansions, a numeration system built in analogy with Rényi βexpansions, was introduced in 2009 by Ito and Sadahiro (see [8]). They gave an alternate-lexicographic criterion for deciding whether some digit string is the (−β)expansion of some x and also described some properties of the corresponding (−β)shift, which was also subject of [2], [3] and [7].…”

Isomorphisms between positive and negative -transformations

“…Recent results in theoretical computer science have shown that negative β-expansions, i.e., expansions of the form x = ∞ k=1 (b k /(−β) k ), have better robustness properties when used for encoding purposes [AHK08]. In [IS09] Ito and Sadahiro studied 154…”

“…Piecewise linear expanding interval maps with constant negative slope in general and the map S I S in particular have also generated a lot of attention. The papers [DK11,Gór09,Hof12,IS09,LS12] deal, among other things, with invariant measures, transitivity and attractors of the maps in question. It seems natural to ask exactly how different the positive and negative slope maps really are.…”

Digital expansions with negative real bases

“…This changed only in recent years, after Ito and Sadahiro [IS09] , i.e., b n = ε n (x) for all n ≥ 1. Since the map x → −βx is order-reversing, the (−β)-admissible sequences are characterized using the alternating lexicographic order.…”

“…Since the map x → −βx is order-reversing, the (−β)-admissible sequences are characterized using the alternating lexicographic order. By [IS09], a sequence b 1 b 2 · · · is (−β)-admissible if and only if…”