Beta-expansions, natural extensions and multiple tilings associated with Pisot units

TL;DR

This paper explores how various transformations related to Pisot units generate multiple tilings of Euclidean space, extending known results and identifying conditions under which tilings occur, with implications for the Pisot conjecture.

Contribution

It introduces new transformations for Pisot units, establishes criteria for tiling properties, and analyzes the symmetric $eta$-transformation's limitations, broadening understanding of $eta$-expansions.

Findings

Certain transformations produce multiple tilings under mild conditions

A necessary and sufficient condition for tiling generalizes the weak finiteness property

Symmetric $eta$-transformation does not satisfy the tiling condition for specific Pisot numbers

Abstract

From the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit $\beta$ and the greedy $\beta$-transformation. In this paper, we consider different transformations generating expansions in base $\beta$, including cases where the associated subshift is not sofic. Under certain mild conditions, we show that they give multiple tilings. We also give a necessary and sufficient condition for the tiling property, generalizing the weak finiteness property (W) for greedy $\beta$-expansions. Remarkably, the symmetric $\beta$-transformation does not satisfy this condition when $\beta$ is the smallest Pisot number or the Tribonacci number. This means that the Pisot conjecture on tilings cannot be extended to the symmetric $\beta$-transformation. Closely related to these (multiple) tilings are natural extensions of the transformations, which have many nice properties: they are invariant under the Lebesgue measure; under certain conditions, they provide Markov partitions of the torus; they characterize the numbers with purely periodic expansion, and they allow determining any digit in an expansion without knowing the other digits.