Geodesic
A Dynamical Proof of Pisot's Theorem
Canadian mathematical bulletin, Tome 49 (2006) no. 1, pp. 108-112

Voir la notice de l'article provenant de la source Cambridge University Press

We give a geometric proof of classical results that characterize Pisot numbers as algebraic $\text{ }\lambda \,>1$ for which there is $x\ne 0$ with $\text{ }\lambda {{\text{ }}^{n}}x\to 0\left( \,\bmod \,\,1 \right)$ and identify such $x$ as members of $\mathbb{Z}\left[ \text{ }\lambda {{\text{ }}^{-1}} \right]\cdot$ $\mathbb{Z}{{\left[ \text{ }\!\!\lambda\!\!\text{ } \right]}^{*}}$ where $\mathbb{Z}{{\left[ \text{ }\!\!\lambda\!\!\text{ } \right]}^{*}}$ is the dual module of $\mathbb{Z}\left[ \text{ }\!\!\lambda\!\!\text{ } \right]$ .
DOI : 10.4153/CMB-2006-010-9
Mots-clés : 11R06 
Kwapisz, Jaroslaw. A Dynamical Proof of Pisot's Theorem. Canadian mathematical bulletin, Tome 49 (2006) no. 1, pp. 108-112. doi: 10.4153/CMB-2006-010-9
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[1] [1] Barge, M. and Kwapisz, J., Geometric theory of Pisot substitutions. In preparation; available at http://www.math.montana.edu/~jarek/papers.html. Google Scholar

[2] [2] Bertin, M.-J., Decomps-Guilloux, A., Grandet-Hugot, M., Pathiaux-Delefosse, M., and Schreiber, J.-P., Pisot and Salem Numbers. Birkhäuser Verlag, Basel, 1992. Google Scholar

[3] [3] Cassels, J. W. S., An Introduction to Diophantine Approximation. Cambridge Tracts in Mathematics and Mathematical Physics 45, Cambridge University Press, NY, 1957. Google Scholar

[4] [4] Pisot, C., La répartition modulo 1 et les nombres algébriques. Ann. Scu. Norm. Sup. Pisa 27(1938), 205–248. Google Scholar

[5] [5] Salem, R., Algebraic numbers and Fourier analysis. D. C. Heath, Boston, MA, 1963. Google Scholar

[6] [6] Solomyak, B., Dynamics of self-similar tilings. Ergodic Theory Dynam. Systems 17(1997), 695–738. Google Scholar

[7] [7] Vijayaraghavan, T., On the fractional parts of the powers of a number. II. Proc. Cambridge Philos. Soc. 37(1941), 349–357. Google Scholar

[8] [8] Weiss, E., Algebraic Number Theory. Chelsea Publishing, New York, NY, 1976. Google Scholar

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