Homological Pisot Substitutions and Exact Regularity

Details Homological Pisot Substitutions and Exact Regularity Homological Pisot Substitutions and Exact Regularity

2010-01-12

arxiv.org/abs/1001.2027v1

Mathematics

Dynamical Systems

16 pages, LaTeX, no figures

Scientific paper

We consider one-dimensional substitution tiling spaces where the dilatation (stretching factor) is a degree d Pisot number, and where the first rational Cech cohomology is d-dimensional. We construct examples of such "homological Pisot" substitutions that do not have pure discrete spectra. These examples are not unimodular, and we conjecture that the coincidence rank must always divide a power of the norm of the dilatation. To support this conjecture, we show that homological Pisot substitutions exhibit an Exact Regularity Property (ERP), in which the number of occurrences of a patch for a return length is governed strictly by the length. The ERP puts strong constraints on the measure of any cylinder set in the corresponding tiling space.

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