Quasimorphismes sur le monoïde libre, et substitutions dans les mesures invariantes

Bousch, Thierry

Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2651-2678.

On montre comment un morphisme $\sigma :A^{☆}\to B^{☆}$ entre deux monoïdes libres de type fini peut transformer une mesure invariante sur $A^{\mathbb{N}}$ en une mesure invariante sur $B^{\mathbb{N}}$. L’existence de cette opération est intimement liée à un résultat de représentation des quasimorphismes homogènes sur le monoïde libre. On étudiera ses propriétés, en particulier vis-à-vis de la distance de réarrangement.

It will be shown how a morphism $\sigma :A^{☆}\to B^{☆}$ between two finitely generated free monoids can transform an invariant mesure on $A^{\mathbb{N}}$ into an invariant measure on $B^{\mathbb{N}}$. The existence of this operation is intimately related to a representation result for homogeneous quasimorphisms on the free monoid. Its properties will be studied, in particular with respect to the rearrangement distance.

Classification : 20M50, 37B10
Mot clés : quasimorphisme, substitution, norme de réarrangement
Keywords: quasimorphism, substitution, rearrangement norm

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