Self-similar tilings

A locally finite tiling of {dollar}IRsp{lcub}n{rcub}{dollar} with a finite number of compact tile shapes is called a self-similar tiling (SST) if there is a linear expanding map {dollar}varphi{dollar} of {dollar}IRsp{lcub}n{rcub}{dollar} such that the image of a tile is the finite union of tiles in the original tiling. In this thesis we discuss several consequences of this property.; We characterize periodic self-similar tilings up to an equivalence (choosing coset representative) as expanding elements of {dollar}glsb{lcub}n{rcub}{dollar}({dollar}doubz{dollar}). A periodic self-similar tiling gives a representation of elements of {dollar}IRsp{lcub}n{rcub}{dollar} in base {dollar}varphi{dollar} with a set of digits D depending on the tiling. We give an algorithm for determining whether or not for a given integer (resp. Gaussian integer) b and set of digits D, every real number (resp. complex number) can be represented in base b with digits from D. In particular we show that in the real case, if D is not projectively rational there are always reals which cannot be so represented.; A general (not necessarily periodic) SST with expansion {dollar}varphi{dollar} gives a representation of numbers in base {dollar}varphi{dollar}, with allowed digit sequences represented by a regular language.; We extend the classification due to Thurston of the expansion constants of general plane SSTs as complex Perron numbers, to tilings of {dollar}IRsp{lcub}n{rcub}{dollar}: a linear map {dollar}varphi{dollar} is the expansion for a SST of {dollar}IRsp{lcub}n{rcub}{dollar} if and only if the eigenvalues of {dollar}varphi{dollar} are algebraic integers and the multiplicity of an eigenvalue {dollar}lambda{dollar} is less than or equal to the multiplicity (as eigenvalues) of all its Galois conjugates which are greater or equal to {dollar}{lcub}mid{rcub}lambda{lcub}mid{rcub}{dollar} in modulus.; We give several constructions of SSTs, most importantly a construction using a wrinkling map on the edges in a polygonal tiling which is approximately self-similar.; As an application we construct Markov partitions for hyperbolic linear maps of the n-torus from SSTs of the unstable manifolds.