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Tilings associated with endomorphisms of free groups
by
E. Arthur (Robbie) Robinson
George Washington University
Coauthors: Shunji Ito (Tsuda College), Maki Furukado (Tsuda College)

A tiling x of R² is called a self-affine if there is an expanding linear mapping B on R² such that each tile in x is a decomposition of tiles in Bx. Let A be an d×d nonsingular hyperbolic integer matrix, let W denote the expanding subspace of A and let B = A|_W. Our goal is to construct a self-affine tiling of W using the method of Dekking.

Let w be an endomorphism of F<d> such that A is the abelianization of w. We view w as a substitution. Note that many substitutions correspond to each A. Let p_i, i=1, ..., d, denote the projections of the standard basis in R^d to W. For i ≠ j, B^-n(wⁿ [p_i, p_j]) is a closed curve which converges to a compact set in the Hausdorff topology. The goal is to choose \theta so that B^-n(wⁿ[p_i, p_j]) is a simple closed curve for each n. When this happens, the method of Dekking yields a Jordan curve and by standard methods in the theory of tilings, we can construct a self-affine tiling corresponding to B.

We concentrate on the case where A is a 4×4 matrix satisfying the non-Pisot condition (two eigenvalues on either side of the unit circle). In this case we compute a 6×6 matrix A^*' and if A^* ≥ 0 we have an algorithm to find a solution. The case A^*\not ≥ 0 remains mysterious.

Date received: December 4, 2000

Copyright © 2000 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cafy-09.