Inflation Species of Planar Tilings Which Are Not of Locally Finite Complexity

L. Danzer

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Discrete geometry and geometry of numbers, Tome 239 (2002), pp. 118-126

Résumé

Let $\mathbf S$ be an inflation species in $\mathbb E^2$ with an inflation factor $\eta$. The following cases are possible: (1) $\mathbf S$ is face-to-face. Then, trivially, there are only finitely many clusters in $\mathbf S$ that fit into a circle of radius $R$, where $R$ is the maximum of the diameters of the prototiles. This property is called locally finite complexity (LFC). If a species is repetitive, it is necessarily in (LFC). (2) $\mathbf S$ is not face-to-face, but $\eta$ is a PV-number. The only class of examples of this type known to the author was published by R. Kenyon in 1992. (3) $\mathbf S$ is not face-to-face and $\eta$ is not a PV-number. For this case, a criterion will be presented that says the following: If, after a finite number of steps, a certain inequality issatisfied, then $\mathbf S$ is not in (LFC) (and, hence, cannot be repetitive). It seems that this is a generic subcase of case (3). In other words, in case (3) (LFC)-species are very rare. No inflation species is known that is not face-to-face with inflation factor $\eta$ not being a PV-number but which is nevertheless in (LFC).