An one-dimensional substitution rule that uses an infinite number of prototiles.
The inflation factor is $2.5$
.
The substitution rules are given by:
$T_{0}\rightarrow T_{0},T_{1}$
$T_{1}\rightarrow T_{0},T_{0},T_{2}$
$T_{2}\rightarrow T_{0},T_{1},T_{3}$
$T_{k}\rightarrow T_{0},T_{k-1},T_{k+1}$
$T_{\infty}\rightarrow T_{0},T_{\infty},T_{\infty}$
The corresponding substitution matrix can be written as:
$1 1 1 1 1 1 1 1 1 … $
$1 0 1 0 0 0 0 0 0 … $
$0 1 0 1 0 0 0 0 0 … $
$0 0 1 0 1 0 0 0 0 … $
$0 0 0 1 0 1 0 0 0 … $
$0 0 0 0 1 0 1 0 0 … $
$0 0 0 0 0 1 0 1 0 … $
$0 0 0 0 0 0 1 0 1 …$
$0 0 0 0 0 0 0 1 0 …$
… … …
The lengths of the proto tiles are given by:
$length(T_{0})=1$
$length(T_{1})=\frac{3}{2}$
$length(T_{2})=\frac{7}{4}$
$length(T_{k})=\frac{2^{k+1}-1}{2^{k}}$
$length(T_{\infty})=2$
[MRW2021]
Manibo, N. and Rust, D. and Walton J. J.},
Spectral properties of substitutions on compact alphabets
arXiv
2021,
https://arxiv.org/abs/2204.07516