Expanding Polynomials and Connectedness of Self-Affine Tiles |
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Authors: | Ibrahim Kirat Ka-Sing Lau and Hui Rao |
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Institution: | (1) Department of Mathematics, Istanbul Technical University, 34469 Maslak-Istanbul, Turkey;(2) Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong;(3) Department of Mathematics, Wuhan University, Wuhan, Peoples Republic of China |
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Abstract: | Little is known about the connectedness of self-affine tiles in
${\Bbb R}^n$. In this note we consider this property on the self-affine tiles
that are generated by consecutive collinear digit sets. By using an algebraic
criterion, we call it the {\it height reducing property}, on expanding polynomials
(i.e., all the roots have moduli $ > 1$), we show that all such tiles in ${\Bbb
R}^n, n \leq 3$, are connected. The problem is still unsolved for higher
dimensions. For this we make another investigation on this algebraic criterion.
We improve a result of Garsia concerning the heights of expanding polynomials.
The new result has its own interest from an algebraic point of view and also
gives further insight to the connectedness problem. |
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Keywords: | |
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