## Dual

One way to compute the window of a model set (resp. the Rauzy fractal of a tiling) is to consider the ‘lifted’ versions of the expanding maps of the substitution (which are automorphisms of the high dimensional lattice `$L$`

) and their counterparts in the internal space. These are contracting maps yielding an iterated function system. The latter is known to have a unique compact nonempty solution. Multiplying the whole iterated function system by an appropriate factor yields the dual substitution. This was outlined in [Thu89]
, see also [Gel97]
, [Fre05]
.
Actually, this is not the only notion of dual tiling. Equivalent concepts in arbitrary dimension are the natural decomposition method [SW02]
and - almost sure - the dual substitutions in [SAI01]
. In dimension one there are even more equivalent concepts, see for instance [Lam98]
.