The linear map that gives the scaling for a substitution rule, before the replacement by new tiles. Often, the linear map is just a scaling by a real number, or, in the plane case - where $\mathbb{R}^2$ is identified with the complex plane - a multiplication by a complex number. Then this number is called the inflation factor.
The algebraic properties of such a number are of special interest. If the tiling is volume hierarchic up to translation, then the inflation factor is an algebraic integer. If it is real, then this is a simple consequence from the fact that the substitution matrix is a nonnegative integer matrix: The largest eigenvalue (unique, real, positive because of the theorem of Perron Frobenius) is the inflation factor to the dimension.
Thurston stated in [Thu89]
that each inflation factor $q$ of a volume hierarchic tiling is a Perron number. That means, $q$ is an algebraic integer which is strictly larger than its algebraic conjugates in modulus (except the complex conjugate of $q$, if $q$ is a complex number). Kenyon gave a proof of the converse (for each complex Perron number there is a volume hierarchic tiling) in [Ken96]
.
An important property of the inflation factor $q$ of a volume hierarchic tiling is, whether it is a PV number or not. If so - and if $q$ is irrational - then this is a strong hint that the tiling is a cut and project tiling. If $q$ is also an algebraic unit, then the internal space will be Euclidean. If not, the internal space may be $p$-adic or a product of Euclidean and $p$-adic spaces. A lot of literature is devoted to this fact. We give just a short and highly incomplete list which may serve as starting points for further reading: [Fog02]
, [Moo00]
, [Lag96]
, [Sir02]
.
[Sir02]
Sirvent, V F, Solomyak, B
Pure discrete spectrum for one-dimensional substitution systems of Pisot type
Canad. Math. Bull.
2002,
45, 4,
pp. 697--710,
MR1941235
[Lag96]
Lagarias, J C
Meyer's concept of quasicrystal and quasiregular sets.
Comm. Math. Phys.
1996,
179, 2,
pp. 365-376,
MR1400744
[Moo00]
Moody, R V
Model sets: A Survey
From Quasicrystals to More Complex Systems
2000,
Axel F, Dénoyer F, and Gazeau J P, Centre de physique Les Houches, Springer,
[Fog02]
N. Pytheas Fogg
Substitutions in Dynamics, Arithmetics and Combinatorics
Lecture Notes in Mathematics,Springer, Berlin
2002,
1794,
[Thu89]
Thurston, William P
Groups, tilings and finite state automata
Amer. Math. Soc. Colloq. Lectures
1989,
[Ken96]
Kenyon, Richard
The construction of self-similar tilings
Geom. Funct. Anal.
1996,
6, 3,
pp. 471--488,
MR1392326