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Pure discrete spectrum for one-dimensional substitution systems of Pisot type. (English) Zbl 1038.37008
A substitution subshift is one generated by the sequence obtained as the limit of images of the string “1” via iterates of a substitution rule \(\zeta:\mathcal A\to\mathcal A^*\), where \(\mathcal A = \{1,2,\dots,d\}\) is called alphabet, \(\mathcal A^* = \bigcup_{n\geq 1} \mathcal A^n\), and \(\zeta(1) = 1B\), \(B\in\mathcal A^*\). The substitution \(\zeta\) is of Pisot type if the Perron-Frobenius eigenvalue \(\lambda\) of the associated matrix \(M_\zeta\) (with entries \(m_{i,j}\) equal to the number of occurrences of the symbol \(i\) in \(\zeta(j)\)) is a Pisot number and the characteristic polynomial of \(M_\zeta\) is irreducible; in other words, all eigenvalues other than \(\lambda\) are of modulus strictly less than one.
There is a long standing conjecture that every substitution subshift of Pisot type has a pure discrete spectrum. This conjecture has been proved true by M. Barge and B. Diamond [Bull. Soc. Math. Fr. 130, 619–626 (2002; Zbl 1028.37008)] for the case of two symbols.
Presented (with a new proof) is a version of so-called “balanced pairs algorithm” of A. N. Livshits [Russ. Math. Surv. 42, 222–223 (1987; Zbl 0648.47004)], allowing one to test for a pure discrete spectrum in substitution subshifts.
Then, the authors provide conditions for a pure discrete spectrum of the tiling flow associated to a substitution subshift of Pisot type, obtained as the suspension over the subshift with the height function constant on each zero-coordinate cylinder and equal to the corresponding coordinate of the Perron-Frobenius eigenvector. They employ the “overlap algorithm” introduced formerly by the second author for two-dimensional tilings [Ergodic Theory Dyn. Syst. 17, 695–738 (1997; Zbl 0884.58062)], and prove an equivalence between certain behavior of this algorithm and a certain behavior of the balanced pairs algorithm. As result, they deduce that pure point spectrum of the tiling flow implies pure point spectrum of the substitution subshift, and that, in the two-symbol case, the tiling flow has a pure point spectrum (in particular, covering the result of Barge and Diamond).
It must be noted (as the authors do), that a stronger result has been recently proved by A. Clark and L. Sadun [Ergodic Theory Dyn. Syst. 23, 1043–1057 (2003; Zbl 1042.37008)] for substitutions of Pisot type: pure point spectrum of the flow is equivalent to pure point spectrum of the subshift.
The paper is concluded with an appendix containing four examples with the balanced pairs algorithm conducted using MATHEMATICA (and testing positive for a pure point spectrum).

37A30 Ergodic theorems, spectral theory, Markov operators
52C23 Quasicrystals and aperiodic tilings in discrete geometry
37B10 Symbolic dynamics
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