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, DÃ©noyer 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