On orbit closure decompositions of tiling spaces by the generalized projection method (Q5931520)

Let \(J(E)\) be the tiling space on \(p\)-dimensional subspace \(E\) of \(R^d\) with a fixed lattice \(L\) by the generalized projection method. The aim of this paper is to show the following theorem: Let \(p':E^\perp\to \text{span}(L^*\cap E^\perp)\) be the orthogonal projection. Define \(p:L\to\text{span}(L^*\cap E^\perp)\) by \(p=p'\circ (\pi^\perp|_L)\). Take a basis \(x_1,\dots,x_k\) of any direct summand \(K\) such that \(L=p^{-1}(\{0\})\oplus K\). Then \(J(E)\) decomposes into a \(k\) parameter family of orbit closures NEWLINE\[NEWLINE\overline{\text{Orb} T(t_1x_1+\cdots+t_k x_k)}NEWLINE\]NEWLINE for \(t_1,\dots,t_k\in R\).NEWLINENEWLINENEWLINEHere \(\pi:R^d\to E\), \(\pi^\perp:R^d\to E^\perp\) are orthogonal projections, \(T(x)\) is a tiling on a \(p\)-dimensional subspace \(E\) of \(R^d\) by the generalized projection method, \(L^*\) is the dual lattice.