# ACAC Seminar Abstract

## Pisot Substitutions, Numeration and Tilings

Speaker: Milton Minervino
Date, Time: Tue, 28 Feb 2012 15:00

Let \sigma be a substitution, i.e., an endomorphism of the free monoid A^*, where A is a finite alphabet. We can associate to every substitution an incidence matrix M_{\sigma}, with entries (M_{\sigma})_ij = |\sigma(j)|_i, for i, j \in A. If the char- acteristic polynomial of M_{\sigma} is irreducible and has a dominant root \alpha which is a Pisot number, we call the associated substitution \sigma a Pisot substitution. Pisot substitutions can be unimodular or non-unimodular, depending on whether \alpha is a unit or not.

Substitutions generate numeration systems (the Dumont-Thomas numeration system), which in particular generalize \beta-expansions. We can expand every non-negative real number in base \alpha and with digits depending on the prefix-suffix automaton associated to the substitution. The \sigma-integers, i.e., all those x \in \R_+ with expansion in which only non-negative powers of \alpha occur, have particular importance.

The main aim is to understand the behaviour of the symbolic dynamical system (X\{sigma}, S) generated by a substitution \sigma, where X_{\sigma} = {S^n u | n \in \N}, u \in A^{\N} is a fixed point of \sigma and S is the shift. In order to accomplish this, a geometrical interpretation of the substitution \sigma is given. In the unimodular case, it consists in finding a suitable Euclidean representation space in which we project the \sigma-integers. Taking the closure of this set we obtain a self-similar compact set with several interesting properties: the Rauzy fractal, or central tile.

In the non-unimodular case, we have to enlarge the representation space adding some non-Archimedean completions of the number field Q(\alpha), in order to get good measure properties of the central tiles. We will show that every non-unimodular Pisot substitution \sigma induces, via this geometrical interpretation, an aperiodic multiple tiling of its representation space, and we provide some tiling conditions.

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