The Pisot conjecture for -substitutions

We prove the Pisot conjecture for $\unicode[STIX]{x1D6FD}$ -substitutions: if $\unicode[STIX]{x1D6FD}$ is a Pisot number, then the tiling dynamical system $(\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D6FD}}},\mathbb{R})$ associated with the $\unicode[STIX]{x1D6FD}$ -substitutio...

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Veröffentlicht in:Ergodic theory and dynamical systems Jg. 38; H. 2; S. 444 - 472
1. Verfasser: BARGE, MARCY
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Cambridge Cambridge University Press 01.04.2018
Schlagworte:
Arithmetic coding
Automorphisms
Tiling
ISSN:0143-3857, 1469-4417
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Zusammenfassung:We prove the Pisot conjecture for $\unicode[STIX]{x1D6FD}$ -substitutions: if $\unicode[STIX]{x1D6FD}$ is a Pisot number, then the tiling dynamical system $(\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D6FD}}},\mathbb{R})$ associated with the $\unicode[STIX]{x1D6FD}$ -substitution has pure discrete spectrum. As corollaries: (1) arithmetical coding of the hyperbolic solenoidal automorphism associated with the companion matrix of the minimal polynomial of any Pisot number is almost everywhere one-to-one; and (2) all Pisot numbers are weakly finitary.
Bibliographie:ObjectType-Article-1
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ISSN:0143-3857
1469-4417
DOI:10.1017/etds.2016.44