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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.
We prove that maps with on a compact surface have symbolic extensions, i.e., topological extensions which are subshifts over a finite alphabet. More precisely we give a sharp upper bound on the so-called symbolic extension entropy, which is the infimum of the topological entropies of all the symbolic extensions. This answers positively a conjecture of S. Newhouse and T. Downarowicz in dimension two and improves a previous result of the author [11].
On donne une condition combinatoire effective suffisante pour que le sytème dynamique
associé à une substitution de type Pisot ait un spectre purement discret. Dans le cas
unimodulaire, cette condition est nécessaire dès que la substitution n'a qu'un cobord
trivial ; elle est vérifiée si et seulement si le fractal de Rauzy associé à la
substitution engendre un pavage auto-similaire et périodique. On en déduit des conditions
de connexité des fractals de Rauzy.
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