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) \fbox{((I) and (R)) $\Longrightarrow$\ ($\ell$R)} (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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