Construction du cœur compact d’un arbre réel par substitution d’arbre  [ Constructing the compact heart of an -tree using a tree substitution ]
Annales de l'Institut Fourier, Volume 61 (2011) no. 3, p. 851-904

Let σ be an automorphism of the free group. Using a train-track representative of its inverse, one can construct the repelling tree T of σ. The free group acts on T by isometries. The dynamical system generated by σ can be interpreted geometrically by the action of the free group restricted to a compact subset of the metric completion of T. This article gives a construction of this subset on a class of examples by introducing tree substitutions. We will insist on the connections between the construction using a tree substitution and the initial symbolic dynamical system.

Étant donné un automorphisme σ d’un groupe libre et un représentant topologique train-track de son inverse, on peut construire un arbre réel T appelé arbre répulsif de σ. Le groupe libre agit sur T par isométries. La dynamique engendrée par σ peut être représentée par l’action du groupe libre restreinte à un sous-ensemble compact bien choisi du complété métrique de T. Cet article construit ce sous-ensemble sur une classe d’exemples en introduisant des opérations appelées substitutions d’arbre ; on mettra en évidence les relations entre la construction par substitution d’arbre et la dynamique symbolique sous-jacente.

DOI : https://doi.org/10.5802/aif.2631
Classification:  37B10,  20E08,  20F65
Keywords: tree substitution, symbolic dynamics, substitution, word combinatorics, free group automorphism, real tree, system of isometries
@article{AIF_2011__61_3_851_0,
     author = {Jullian, Yann},
     title = {Construction du c\oe ur compact d'un arbre r\'eel par substitution d'arbre},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {3},
     year = {2011},
     pages = {851-904},
     doi = {10.5802/aif.2631},
     mrnumber = {2918720},
     zbl = {1277.37022},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2011__61_3_851_0}
}
Jullian, Yann. Construction du cœur compact d’un arbre réel par substitution d’arbre. Annales de l'Institut Fourier, Volume 61 (2011) no. 3, pp. 851-904. doi : 10.5802/aif.2631. http://www.numdam.org/item/AIF_2011__61_3_851_0/

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