Substitutions: Exercises II
CIMPA School, Isfahan
N. Bédaride ∗
The questions with z are much more difficult. Do not start with them.
Exercice 1. Consider the following substitution where the inflation maps are different
for each square.
2 4
2
1
1 3
3
1
2
1 3
4
1
— Draw some iterations of 1
— Consider the lines and the rows of the tiling. Do they belong to the susbshift defined
by a 1d substitution ?
— Compute the incidence matrix of the substitution. What is the link with an incidence matrix of some 1d substitution ?
Exercice 2. We consider a susbtitutive tiling of the line.
— Consider two intervals I1 , I2 in R. Find their lengths such that a substitution maps
I1 on an interval tiled by I1 , I2 and I2 on an interval translated of I1 .
— What do you remark ?
— zProve that the action of R on this tiling is the suspension of the Z action on the
subshift.
√
Exercice 3. Consider a right triangle of edges 1, 2, 5. Then we define a map by the
rule
1. Draw some iterations of the map.
2. Prove that a tile appears in an infinite number of orientations.
3. Is this map a substitution ?
∗
Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France. Email:
[email protected]
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