Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

A self-affine tile is a compact set $T$ in $\mathbb R^n$ of positive Lebesgue measure for which there is an $n\times n$ expanding matrix $A$ (i.e. all its eigenvalues have modulus greater than 1) such that the affinely inflated copy $A(T)$ of $T$ can be perfectly tiled with essentially disjoint translates of $T$.

Thus we have $$ A(T) = \cup_{i=1}^m (T+d_i); \mathcal D= d_1,d_2,\dots,d_m $$

where $|det(A)| =|\mathcal D|= m$

Results of Kenyon (Projecting the one-dimensional Sierpinski gasket Projecting the one-dimensional Sierpinski gasket. Israel J. Math. 97 (1997), 221--238.) and Lagarias and Wang (Self-affine tiles in $R^n$. Adv. Math. 121 (1996), no. 1, 21--49) tells that such sets always can be used to give a translational tiling of $R^n$ and has boundary of measure zero and has nonempty interiors.

Thus in one dimension we can think of them as a union of intervals (possibly infinitely many ).

My question is :-

Is there a characterization of self-affine tiles in $\mathbb R$ which are union of finitely many intervals ?

share|improve this question
2  
I know that this is not necessary to state the question, but why don't you add some definitions, perhaps references to articles you have already looked at or even a little motivation? –  Gjergji Zaimi Sep 9 '10 at 6:35
2  
I mostly agree with Gjergji, except that I think that it is not only advisable but definitely necessary to state the definitions, so that the question makes sense. –  Victor Protsak Sep 9 '10 at 7:02
1  
Are self-affine tiles a generalisation of rep-tiles? Are they the same in dimension 1? –  Henry Segerman Sep 9 '10 at 7:15
1  
In the paper of Kenyon mentioned above he showed that all prototiles which can be used to give a self-replicating tiling are necessarily self-affine tiles. On the other hand Theorem 1.2 of the Lagarias wang paper says any self-affine tile can be used as a prototile to give a self-replicating tiling of $\mathbb R^n$ –  Vagabond Sep 9 '10 at 7:31
    
@ Gjergji Zaimi and Victor Prostak made the changes as you have recommended. Please let me know if I should add more details. –  Vagabond Sep 9 '10 at 7:33
show 1 more comment

1 Answer

up vote 2 down vote accepted

The classification is given in section 5 of "Integral Self-Affine Tiles in $\mathbb R^n$ I. Standard and Nonstandard Digit Sets" by Lagarias and Wang (Theorem 5.2 and corollary 5.2a). Their result builds on the previous paper by A. M. Odlyzko, "Non-negative digit sets in positional number systems".

share|improve this answer
    
Is there a classification of tiling sets associated with a self affine set which gives a self-replicating tiling of $R^n$ ? Is there some way to uniquely associate such a tiling set with a self-affine set ( I am asking about self-replicating tiling ofcourse). Can you suggest some text/ review paper where I can find a good updated survey of self replicating tiling by self-affine sets. Specially the number theoretic aspects ? –  Vagabond Sep 9 '10 at 8:37
1  
I'm not sure, but if you refer to results of tilings up to $\mathbb Z$ similarity then some work has been done, see for example springerlink.com/content/euyjt5a162b0nt9u –  Gjergji Zaimi Sep 9 '10 at 8:55
1  
You may find useful J.P. Gabardo, X.Yu, Natural tiling, lattice tiling and Lebesgue measure of integral self-affine tiles, J. Lond. Math. Soc., II. Ser. 74 (1) (2006) 184-204. –  Ievgen Bondarenko Sep 9 '10 at 9:15
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.