Proximality in Pisot tiling spaces

Marcy Barge¹ ; Beverly Diamond²

¹ Department of Mathematics Montana State University Bozeman, MT 59717, U.S.A.
² Department of Mathematics College of Charleston Charleston, SC 29424, U.S.A.

Fundamenta Mathematicae, Tome 194 (2007) no. 3, pp. 191-238

Résumé

A substitution $\varphi$ is strong Pisot if its abelianization matrix is nonsingular and all eigenvalues except the Perron–Frobenius eigenvalue have modulus less than one. For strong Pisot $\varphi$ that satisfies a no cycle condition and for which the translation flow on the tiling space ${\cal T}_\varphi$ has pure discrete spectrum, we describe the collection ${\cal T}^{\rm P}_\varphi$ of pairs of proximal tilings in ${\cal T}_\varphi$ in a natural way as a substitution tiling space. We show that if $\psi$ is another such substitution, then ${\cal T}_\varphi $ and ${\cal T}_\psi$ are homeomorphic if and only if ${\cal T}^{\rm P}_\varphi$ and ${\cal T}^{\rm P}_\psi$ are homeomorphic. We make use of this invariant to distinguish tiling spaces for which other known invariants are ineffective. In addition, we show that for strong Pisot substitutions, pure discrete spectrum of the flow on the associated tiling space is equivalent to proximality being a closed relation on the tiling space.

Détail

DOI : 10.4064/fm194-3-1

Keywords: substitution varphi strong pisot its abelianization matrix nonsingular eigenvalues except perron frobenius eigenvalue have modulus strong pisot varphi satisfies cycle condition which translation flow tiling space cal varphi has pure discrete spectrum describe collection cal varphi pairs proximal tilings cal varphi natural substitution tiling space psi another substitution cal varphi cal psi homeomorphic only cal varphi cal psi homeomorphic make invariant distinguish tiling spaces which other known invariants ineffective addition strong pisot substitutions pure discrete spectrum flow associated tiling space equivalent proximality being closed relation tiling space