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

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).