A combinatorial approach to products of Pisot substitutions. (English) Zbl 1378.37036
Summary: We define a generic algorithmic framework to prove a pure discrete spectrum for the substitutive symbolic dynamical systems associated with some infinite families of Pisot substitutions. We focus on the families obtained as finite products of the three-letter substitutions associated with the multidimensional continued fraction algorithms of Brun and Jacobi-Perron. Our tools consist in a reformulation of some combinatorial criteria (coincidence conditions), in terms of properties of discrete plane generation using multidimensional (dual) substitutions. We also deduce some topological and dynamical properties of the Rauzy fractals, of the underlying symbolic dynamical systems, as well as some number-theoretical properties of the associated Pisot numbers.
MSC:
| 37B50 | Multi-dimensional shifts of finite type, tiling dynamics (MSC2010) |
| 37B10 | Symbolic dynamics |
| 28A80 | Fractals |
| 37A45 | Relations of ergodic theory with number theory and harmonic analysis (MSC2010) |
| 11K16 | Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. |
| 52C22 | Tilings in \(n\) dimensions (aspects of discrete geometry) |
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