Symbolic discrepancy and self-similar dynamics

Adamczewski, Boris

Annales de l'Institut Fourier, Tome 54 (2004) no. 7, pp. 2201-2234.

Nous considérons des systèmes dynamiques naturellement associés aux substitutions primitives et connus pour être uniquement ergodiques. Afin d'étudier plus précisément cette propriété, nous introduisons différentes notions de discrépance symbolique. Nous montrons comment les propriétés de répartition d'un tel système sont en partie déterminées par les matrices d'incidences associées à la substitution sous-jacente. Nous donnons également certaines applications de ces résultats à l'étude spectrale des systèmes dynamiques substitutifs.

We consider subshifts arising from primitive substitutions, which are known to be uniquely ergodic dynamical systems. In order to precise this point, we introduce a symbolic notion of discrepancy. We show how the distribution of such a subshift is in part ruled by the spectrum of the incidence matrices associated with the underlying substitution. We also give some applications of these results in connection with the spectral study of substitutive dynamical systems.

Classification : 11K38, 37A30, 37A45, 37B10, 68R15
Keywords: Discrepancy, substitutions, subshifts, bounded remainder sets, self-similar dynamics
Mot clés : discrépance, substitutions, sous-shifts, ensembles à restes bornés, dynamiques auto-similaires

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