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Symbolic discrepancy and self-similar dynamics

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Symbolic discrepancy and self-similar dynamics AN N A L E S D E L’INSTI T U T F O U R IE R ANNALES DE L’INSTITUT FOURIER Boris ADAMCZEWSKI Symbolic discrepancy and self-similar dynamics Tome 54, no 7 (2004), p. 2201-2234. <> © Association des Annales de l’institut Fourier, 2004, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (, implique l’accord avec les conditions générales d’utilisation ( Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques 2201- SYMBOLIC DISCREPANCY AND SELF-SIMILAR DYNAMICS by Boris ADAMCZEWSKI 1. Introduction. In this paper, we introduce two functions of discrepancy, one asso- ciated with symbolic sequences and the other with subshifts (respectively defined in (1) and (2)). We mainly deal with the asymptotic behaviour of these two functions, focusing on sequences obtained as fixed points of primitive substitutions and on subshifts arising from them. Such se- quences naturally appear as soon as one studies dynamical systems with a self-similar structure (that is, the induced system on some subset is topo- logically conjugate to the original one). This is in particular the case for one-dimensional toral quadratic rotations (see for instance [1]) and interval exchanges with parameters lying in the same quadratic field [3]. This work is motivated by questions arising at once from Diophantine approximation and ergodic theory and shares some links with [2], [22], [23], [8], [15]. We first recall the definition of dis

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