Tilings Encyclopedia | Kenyon's Construction

Kenyon's Construction

As well as showing that there are substitution rules for every Perron Inflation Factor, [Ken96] gives an explicit construction for expansion factors $x$ with $x^n - p x^n-1+qx + r =0$ , where $n,p,q,r$ are integers with $n>2, r$ positive, $p,q$ non-negative.

Kenyon (1,2,1)

As well as showing that there are substitution rules with any Perron inflation factor, in [Ken96] , R. Kenyon gives an explicit construction for the Perron numbers that satsify: $xn - a xn-1 + b x + c$, where $a, b$, and $c$ are natural numbers. This is an example of that method given in that paper. A locally isomorphic version with polygonal tiles is Kenyon (1,2,1) Polygon.

Finite Rotations Self Similar Substitution Kenyons Construction

Kenyon (1,2,1) Polygon

A polygonal version of Kenyon (1,2,1). The boundary is generated by the morphism $a \to b, b \to c, c \to c a' b' b'$ (where $x'$ is the inverse of $x$).

Finite Rotations Polytopal Windowed Tiling Canonical Substitution Tiling Parallelogram Tiles Kenyons Construction

References

[Ken96]
Kenyon, Richard
The construction of self-similar tilings
Geom. Funct. Anal. 1996, 6, 3, pp. 471--488, MR1392326