Abstract
We study the pure periodicity of β-expansions where β is a Pisot number satisfing the following two conditions: the β-expansion of 1 is equal to k 1 k 2...k d−11, k i ≥ 0, and the minimal polynomial of β is given by x d − k 1 x d−1 − ... − k d−l x − 1. From the substitution associated with the Pisot number β, a domain with a fractal boundary, called atomic surface, is constructed. The essential point of the proof is to define a natural extension of the β-transformation on a d-dimensional product space which consists of the unit interval and the atomic surface.
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© 2002 Springer Science+Business Media Dordrecht
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Ito, S., Sano, Y. (2002). Substitutions, Atomic Surfaces, and Periodic Beta Expansions. In: Jia, C., Matsumoto, K. (eds) Analytic Number Theory. Developments in Mathematics, vol 6. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3621-2_12
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DOI: https://doi.org/10.1007/978-1-4757-3621-2_12
Publisher Name: Springer, Boston, MA
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