15. ÖMG-Kongress
Jahrestagung der Deutschen Mathematikervereinigung

16. bis 22. September 2001 in Wien

Sektion 5 - Geometrie
Dienstag, 18. September 2001, 16.00, Hörsaal 50

 

When are inflation-species linearly repetitive ($\ell$R)?

Ludwig Danzer, Universität Dortumund

 

Terms: Tiles $T_k$, protoset $\cal F$, cluster $\cal C$, $r$-cluster (fits into a ball of radius $r$), species (family of $\cal F$-tilings invariant under isometries), inflation (I), $S(\cal F,{\rm infl})$ a species defined by inflation, (IM) inflation species with primitive inflation matrix.
Properties of species (not only of individual tilings): Locally finite complexity (LFC) := up to isometries there are only finitely many 2-element-clusters; repetitive (weakly, linearly): (wR), (R), ($\ell$R)

Statements:
(1) (LFC) $\Longrightarrow$ for each $r$ there are only finitely many $r$-clusters
(2) ((wR) and (LFC)) $\Longleftrightarrow$ (R)
(3) $\cal P\in S(\cal F, {\rm infl}) \Longrightarrow \exists \cal Q: \cal Q\in S(\cal F,{\rm infl})$ and $\cal P = {\rm infl}(\cal Q)$ (in general $\cal Q$ is not unique)
(4) $M$ primitive $\Longrightarrow \exists n : M^n > 0$; hence $\cal F$ is minimal
(5) (IM) $\Longrightarrow$ (wR); ((I) and (wR)) $\Longrightarrow$ (IM)
(6) ((IM) and (LFC)) $\Longrightarrow$ (R) (combination of (5) and (2))
(7) (no more assumptions needed!)

On the other hand:
(8) (IM) $\not\Longrightarrow$ (LFC) and hence (wR) $\not\Longrightarrow$ (LFC) (cf (5)) and (IM) $\not\Longrightarrow$ (R) (cf (2))
(9) ((I) and (LFC)) $\not\Longrightarrow$ (wR)

$^*$ Everything with respect to isometries. No restriction to translations.

E-Mail:

danzer@math.uni-dortmund

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