Interval exchange transformation extension of a substitution dynamical system
Confluentes Mathematici, Volume 4 (2012) no. 4.

Let α be a free group automorphism on FN with maximal index and Σα the associated attracting subshift. We prove that there exists an Interval Exchange Transformation on the circle whose coding factorizes onto Σα. In the case when there is an explicit construction of the ℝ-tree associated to α, we construct algorithmically such IET. We conclude by giving examples.

Published online:
DOI: 10.1142/S1793744212500053
Xavier Bressaud 1; Yann Jullian 1

1
@article{CML_2012__4_4_A1_0,
     author = {Xavier Bressaud and Yann Jullian},
     title = {Interval exchange transformation extension of a substitution dynamical system},
     journal = {Confluentes Mathematici},
     publisher = {World Scientific Publishing Co Pte Ltd},
     volume = {4},
     number = {4},
     year = {2012},
     doi = {10.1142/S1793744212500053},
     language = {en},
     url = {https://cml.centre-mersenne.org/articles/10.1142/S1793744212500053/}
}
TY  - JOUR
AU  - Xavier Bressaud
AU  - Yann Jullian
TI  - Interval exchange transformation extension of a substitution dynamical system
JO  - Confluentes Mathematici
PY  - 2012
VL  - 4
IS  - 4
PB  - World Scientific Publishing Co Pte Ltd
UR  - https://cml.centre-mersenne.org/articles/10.1142/S1793744212500053/
DO  - 10.1142/S1793744212500053
LA  - en
ID  - CML_2012__4_4_A1_0
ER  - 
%0 Journal Article
%A Xavier Bressaud
%A Yann Jullian
%T Interval exchange transformation extension of a substitution dynamical system
%J Confluentes Mathematici
%D 2012
%V 4
%N 4
%I World Scientific Publishing Co Pte Ltd
%U https://cml.centre-mersenne.org/articles/10.1142/S1793744212500053/
%R 10.1142/S1793744212500053
%G en
%F CML_2012__4_4_A1_0
Xavier Bressaud; Yann Jullian. Interval exchange transformation extension of a substitution dynamical system. Confluentes Mathematici, Volume 4 (2012) no. 4. doi : 10.1142/S1793744212500053. https://cml.centre-mersenne.org/articles/10.1142/S1793744212500053/

[1] P. Arnoux, J. Bernat and X. Bressaud, Geometrical models for substitutions, Exper- imental Math. 20 (2011) 97–127.

[2] P. Arnoux, V. Berthé, A. Hilion and A. Siegel, Fractal representation of the attractive lamination of an automorphism of the free group, Ann. Inst. Fourier 56 (2006) 2161– 2212.

[3] P. Arnoux and S. Ito, Pisot substitutions and rauzy fractals, Bull. Belg. Math. Soc. 8 (2001) 181–207.

[4] P. Arnoux, Un exemple de semi-conjugaison entre un échange d’intervalles et une translation sur le tore, Bull. Soc. Math. France 116 (1988) 489–500.

[5] P. Arnoux and J.-C. Yoccoz, Construction de difféomorphismes pseudo-Anosov, Comp. Acad. Sci., Sér. I 292 (1981) 75–78.

[6] M. Bestvina and M. Feighn, Outer limits, 1994, preprint.

[7] M. Bestvina and M. Handel, Train-tracks and automorphisms of free groups, Ann. Math. 135 (1992) 1–51.

[8] M. Boshernitzan and I. Kornfeld, Interval translation mappings, Ergodic Theory Dynam. Systems 15 (1995) 821–832.

[9] T. Coulbois and A. Hilion, Rips induction: Index of the dual lamination of an R-tree, arXiv:1002.0972.

[10] T. Coulbois and A. Hilion, Botany of irreducible automorphisms of free groups, Pacific J. Math. 256 (2012) 291–307.

[11] T. Coulbois, A. Hilion and M. Lustig, R-trees and laminations for free groups II: The dual lamination of an R-tree, J. London Math. Soc. 78 (2008) 737–754.

[12] T. Coulbois, A. Hilion and M. Lustig, R-trees, dual laminations, and compact systems of partial isometries, Math. Proc. Cambridge Phil. Soc. 147 (2009) 345–368.

[13] M. M. Cohen and M. Lustig, Very small group actions on R-trees and Dehn twist automorphisms, Topol. 34 (1995) 575–617.

[14] M. Culler and J. Morgan, Group actions on R-trees, Proc. London Math. Soc. 55 (1987) 571–604.

[15] T. Coulbois, Fractal trees for irreducible automorphisms of free groups, J. Mod. Dynam. 4 (2010) 359–391.

[16] V. Canterini and A. Siegel, Automate des préfixes-suffixes associé à une substitution primitive, J. Th. Nombres Bordeaux 13 (2001) 353–369.

[17] V. Canterini and A. Siegel, Geometric representation of substitutions of Pisot type, Trans. Amer. Math. Soc. 353 (2001) 5121–5144.

[18] K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Mathematics (Cambridge Univ. Press, 1985).

[19] A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les Surfaces, Astérisque, Vols. 66–67 (Soc. Math. France, 1979).

[20] D. Gaboriau, A. Jaeger, G. Levitt and M. Lustig, An index for counting fixed points of automorphisms of free groups, Duke Math. J. 93 (1998) 425–452.

[21] D. Gaboriau and G. Levitt, The rank of action on R-trees, Ann. Scient. Ec. Norm. Sup. (4) 28 (1995) 549–570.

[22] M. Handel and L. Mosher, Axes in outer space, 2006.

[23] M. Handel and L. Mosher, Parageometric outer automorphisms of free groups, Trans. Amer. Math. Soc. 359 (2007) 3153–3183.

[24] Y. Jullian, Représentation géométrique des systèmes dynamiques substitutifs par sub- stitutions d’arbre, PhD thesis, Université de la Méditerranée, 2009.

[25] Y. Jullian, Construction du cœur compact d’un arbre réel par substitution d’arbre, to appear in Ann. Inst. Fourier, arXiv:1002.3933.

[26] Y. Jullian, Explicit computation of the index of a positive automorphism of the free group, arXiv:1012.3267.

[27] Y. Jullian, Positive automorphisms for interval exchange transformations, arXiv:1107.2430.

[28] I. Kapovich and M. Lustig, Invariant laminations for irreducible automorphisms of free groups, arXiv:1104.1265v2.

[29] G. Levitt and M. Lustig, Irreducible automorphisms of Fn have north–south dynamics on compactified outer-space, J. Inst. Math. Jussieu 2 (2003) 59–72.

[30] R. Daniel Mauldin and S. C. Williams, Hausdorff dimension in graph directed con- structions, Trans. Amer. Math. Soc. 309 (1988) 811–829.

[31] N. Pytheas Fogg, Arithmetics and Combinatorics, Lecture Notes in Mathematics, Vol. 1794, Substitutions in Dynamics (Springer-Verlag, 2002), eds. V. Berthé, S. Fer- enczi, C. Mauduit and A. Siegel.

[32] M. Queffélec, Substitution Dynamical Systems-Spectral Analysis, Lecture Notes in Mathematics, Vol. 1294 (Springer-Verlag, 1987).

[33] G. Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France 110 (1982) 147–178.

[34] M. Viana, Dynamics of Interval Exchange Maps and Teichmüller Flows (IMPA, 2005 and 2007).

[35] K. Vogtmann, Automorphisms of free groups and outer space, Geometriæ Dedicata 94 (2002) 1–31.

[36] J.-C. Yoccoz, Continued fraction algorithms for interval exchange maps: An introduc- tion, in Frontiers in Number Theory, Physics and Geometry 1 (Springer, 2006).

Cited by Sources: