12 1. BASIC NOTIONS Exercise 1.7. Show that for the Thue-Morse substitution, Ωσ = ΩT . (2) The Fibonacci substitution is σ(a) = b, σ(b) = ab. The substitution matrix is ( 0 1 1 1 ) , whose Perron-Frobenius eigenvalue equals the golden mean τ = (1 + √ 5)/2. The corresponding left- and right-eigenvectors are L = (1, τ ) and R = ( 1 τ ) . As with the Thue-Morse sequence, there is a fixed point of σ2 built from the seed b.a. To get a tiling, we take the a-tile to have length 1 and the b-tile to have length τ . On average, there are τ b-tiles for every a-tile. This shows that there are no periodic tilings in Ωσ, since the ratio of a to b-tiles in a periodic tiling would have to be rational. (3) The period-doubling substitution has σ(a) = bb, σ(b) = ab. The matrix is ( 0 1 2 1 ) , with λP F = 2, L = (1, 1) and R = 1 2 . Once again, a fixed point of σ2 can be built from the seed b.a There is a map from the Thue-Morse tiling space to the period- doubling tiling space. If a Thue-Morse tile is preceded by a tile of the same type, replace it with a period-doubling a tile. If it is preceded by a tile of the opposite type, replace it with a period-doubling b tile. Exercise 1.8. Let ΩT M and ΩP D denote the Thue-Morse and period- doubling substitution tiling spaces, respectively, and let f : ΩT M → ΩP D be as above. Show that f intertwines the two substitutions. That is, σP D ◦ f = f ◦ σT M . Use this fact to show that f is a 2:1 cover of ΩP D by ΩT M . A brief digression into history. Substitution sequences were studied at length long before aperiodic tilings became fashionable. Thue invented the Thue-Morse sequence in the late 1800s, and Morse reinvented it in the 1930s to prove properties of geodesics on Riemann surfaces. Traditionally, the central object of study wasn’t the space of sequences, but rather a particular sequence that was fixed by some power of the substitution. The sequence space could then be recovered from this fixed point by taking its orbit under a shift map, and then taking the completion of that orbit in a metric that is similar to our tiling metric. Substitution tilings in higher dimensions. A substitution in one dimension is combinatorial — replace each letter with a word. In higher di- mensions, we must also consider the geometry of the tiles. Given a stretching factor λ 1, a substitution is an operation that (1) Stretches each tile by a linear factor λ, and (2) Replaces each stretched tile by a cluster of (ordinary-sized) tiles, as in Figure 1.10. As before, these clusters are called supertiles (of order 1). The substitution matrix is as before, and Mij gives the number of i-tiles in a substituted j-tile. The left-eigenvector L = (L1, . . . , Lk) specifies their volumes of the different tile types, but has no information about their shapes.

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