TY - BOOK

T1 - Geometry and topology of self-affine tiles and Rauzy fractals

AU - Zhang, Shuqin

N1 - no embargo

PY - 2019

Y1 - 2019

N2 - In this thesis we focus on topological problems related to self-affine tiles, fractal crystallographic tiles and the construction of space-filling curves for self-affine tiles and Rauzy fractals. In the first part we consider the unique solution T=T(M,D) of the set equation MT=T+D in the three dimensional space, where M is a 3 times 3 expanding integer matrix and D in is a digit set containing integer vectors. If T has positive Lebesgue measure it is called a 3-dimensional self-affine tile. If a digit set only consists of multiples of a given nonzero vector v we call it a collinear digit set. We prove that the boundary of a self-affine tile T with collinear digit set is homeomorphic to a two dimensional sphere whenever its set of neighbors in a lattice tiling which is induced by T in a natural way contains 14 elements. Moreover, we give a characterization of 3-dimensional self-affine tiles with collinear digit set having 14 neighbors in terms of the coefficients of the characteristic polynomial of M. In our proofs we use results of R. H. Bing on the topological characterization of spheres. Our approach can be turned into an algorithm that allows to check if a given 3-dimensional self-affine tile with 14 neighbors has spherical boundary and even has the potential to be generalized to higher dimensions. In the second part, we study topological properties of a class of planar crystallographic replication tiles. Let M be an expanding 2 times 2 matrix with characteristic polynomial p(x) and let v be an integer vector such that v and Mv are linearly independent. Then M and v can be used to set up a set equation that defines a unique nonempty compact set T which is equal to the closure of its interior. Moreover, T tiles the plane with respect to the crystallographic group p2 generated by the rotation by 180 degrees and the translations by integer vectors. It was proved by Leung and Lau in the context of self-affine lattice tiles with collinear digit set that the union of T with (-T) is homeomorphic to a closed disk if and only if the coefficients of the characteristic polynomial p(x) satisfiey a certain inequality. However, this characterization does not hold anymore for T itself. In this thesis, we completely characterize the tiles T of this class that are homeomorphic to a closed disk. At the end of the thesis, we make efforts to construct the space-filling curves of self-affine sets and Rauzy fractals. Inspired by the previous systematic work of constructing space-filling curves of self-similar sets by Dai, Rao and Zhang, we generalize the concept of optimal parametrizations to the invariant sets of a single matrix graph directed IFS. We show that the invariant sets of a linear single matrix graph directed IFS which has primitive associated matrix and satisfy the open set condition admit optimal parametrizations. This result is the basis of further study. We can extend the definition of skeleton, ordered graph directed IFS, and linear graph directed IFS to the graph-directed IFS. In terms of these, we are able to systematically construct space-filling curves of self-affine sets and Rauzy fractals.

AB - In this thesis we focus on topological problems related to self-affine tiles, fractal crystallographic tiles and the construction of space-filling curves for self-affine tiles and Rauzy fractals. In the first part we consider the unique solution T=T(M,D) of the set equation MT=T+D in the three dimensional space, where M is a 3 times 3 expanding integer matrix and D in is a digit set containing integer vectors. If T has positive Lebesgue measure it is called a 3-dimensional self-affine tile. If a digit set only consists of multiples of a given nonzero vector v we call it a collinear digit set. We prove that the boundary of a self-affine tile T with collinear digit set is homeomorphic to a two dimensional sphere whenever its set of neighbors in a lattice tiling which is induced by T in a natural way contains 14 elements. Moreover, we give a characterization of 3-dimensional self-affine tiles with collinear digit set having 14 neighbors in terms of the coefficients of the characteristic polynomial of M. In our proofs we use results of R. H. Bing on the topological characterization of spheres. Our approach can be turned into an algorithm that allows to check if a given 3-dimensional self-affine tile with 14 neighbors has spherical boundary and even has the potential to be generalized to higher dimensions. In the second part, we study topological properties of a class of planar crystallographic replication tiles. Let M be an expanding 2 times 2 matrix with characteristic polynomial p(x) and let v be an integer vector such that v and Mv are linearly independent. Then M and v can be used to set up a set equation that defines a unique nonempty compact set T which is equal to the closure of its interior. Moreover, T tiles the plane with respect to the crystallographic group p2 generated by the rotation by 180 degrees and the translations by integer vectors. It was proved by Leung and Lau in the context of self-affine lattice tiles with collinear digit set that the union of T with (-T) is homeomorphic to a closed disk if and only if the coefficients of the characteristic polynomial p(x) satisfiey a certain inequality. However, this characterization does not hold anymore for T itself. In this thesis, we completely characterize the tiles T of this class that are homeomorphic to a closed disk. At the end of the thesis, we make efforts to construct the space-filling curves of self-affine sets and Rauzy fractals. Inspired by the previous systematic work of constructing space-filling curves of self-similar sets by Dai, Rao and Zhang, we generalize the concept of optimal parametrizations to the invariant sets of a single matrix graph directed IFS. We show that the invariant sets of a linear single matrix graph directed IFS which has primitive associated matrix and satisfy the open set condition admit optimal parametrizations. This result is the basis of further study. We can extend the definition of skeleton, ordered graph directed IFS, and linear graph directed IFS to the graph-directed IFS. In terms of these, we are able to systematically construct space-filling curves of self-affine sets and Rauzy fractals.

KW - Self-affine tiles

KW - Crystallographic p2-tiles

KW - Space-filling curves

KW - Rauzy fractals

KW - Neighbor sets

KW - Neighbor graph

KW - Low dimensional topology

KW - Selbstaffine Kachel

KW - Kristallographischer p2-Kacheln

KW - Raumfüllende Kurve

KW - Rauzyfraktale

KW - Nachbarmengen

KW - Nachbargraph

KW - niedrigdimensionale Topologie

M3 - Doctoral Thesis

ER -