I've been trying to understand more on "geometric" substitutions rather than just symbolic ones. As symbolic substitutions always yield FLC tilings, I wanted to know whether a tiling coming from a substitution being FLC, depends on the starting seed?
More accurately, I have a finite prototile set $\mathcal{P}$ and a substitution $S:\mathcal{P}\to \mathcal{P}^*$, where $\mathcal{P}^*$ are valid patches using the prototiles. I then take a finite seed $P_0\in \mathcal{P}^*$ and consider the sequence $P_n:=S^n(P_0)\in \mathcal{P}^*$. Then for any $k\in \mathbb{N}$, is the number of $k$-coronas occuring in $P_n$ uniformly bounded, independent of what is $P_0$?
When dealing with symbolic substitutions over a finite alphabet $\mathcal{A}$ on $\mathbb{Z}^d$, we have a bound of the form $\vert \mathcal{A} \vert^{(2k+1)^d}$ for how many $k$-coronas can appear in $P_n$, regardless of the seed.
In the geometric case, if I start with a legal seed $P_0$, then this doesn't seem like a problem. Moreover, if $\mathcal{T}$ is a tiling which is a limit of $P_n$, the uniform bound should come from the number of $k$-coronas in $\mathcal{T}$. Can it be that a legal tiling $\mathcal{T}$ has finitely many $k$-coronas, but the number of $k$-coronas in $P_n$ keeps growing? Are there conditions for the number of $k$-coronas to be uniformly bounded, regardless of the seed $P_0$?
I hope the terminology I used is coherent and would appreciate any input on this matter.
