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Geometric theory of unimodular Pisot substitutions. (English) Zbl 1152.37011
The authors are concerned with the tiling flow \(T\) associated to a substitution \(\phi\) over a finite alphabet. Their focus is on substitutions that are unimodular Pisot, namely their matrix is unimodular and has all eigenvalues strictly inside the unit circle with exception of the Perron eigenvalue \(\lambda>1\). Thus \(\lambda\) belongs to the class of Pisot numbers. The motivation is provided by the still open conjecture asserting that \(T\) has pure discrete spectrum, including: injectivity of the canonical torus map (the geometric realization), geometric coincidence condition, partial commutation of \(T\) and the dual \(\mathbb R^{d-1}\)-action, measure and tiling properties of Rauzy fractals, and concrete algorithms. Some of these are original and some have already appeared in the literature (as sufficient conditions only) but they all emerge from a unified approach based on the new device: the strand space \(\mathcal F_\phi\) of \(\phi\). The proof of the necessity hinges on determination of the discrete spectrum of \(T\) as that of the associated Kronecker toral flow.

MSC:
37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
37B10 Symbolic dynamics
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
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