Higher dimensional extensions of substitutions and their dual maps |
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Authors: | Yuki Sano Pierre Arnoux Shunji Ito |
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Affiliation: | (1) Department of Mathematics, Tsuda College, Tsuda-Machi, Kodaira, 187 Tokyo, Japan;(2) Institut de Mathématiques de Luminy (UPR 9016), 163 Avenue de Luminy Case 930, 13288 Marseille Cedex 9, France |
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Abstract: | Given a substitution σ ond letters, we define itsk-dimensional extension,E k (σ), for 0≤k≤d. Thek-dimensional extension acts on the set ofk-dimensional faces of unit cubes inR d with integer vertices. The extensions of a substitution satisfy a commutation relation with the natural boundary operator: the boundary of the image is the image of the boundary. We say that a substitution is unimodular (resp. hyperbolic) if the matrix associated to the substitution by abelianization is unimodular (resp. hyperbolic). In the case where the substitution is unimodular, we also define dual substitutions which satisfy a similar coboundary condition. We use these constructions to build self-similar sets on the expanding and contracting space for an hyperbolic substitution. |
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