%0 Journal Article %T Geometrical Models for Substitutions %+ Institut de mathématiques de Luminy (IML) %+ Institut Élie Cartan de Nancy (IECN) %+ Institut de Mathématiques de Toulouse UMR5219 (IMT) %A Arnoux, Pierre %A Bernat, Julien %A Bressaud, Xavier %< avec comité de lecture %@ 1058-6458 %J Experimental Mathematics %I Taylor & Francis %V 20 %N 1 %P 97-127 %8 2011-10-14 %D 2011 %R 10.1080/10586458.2011.544590 %K substitutive dynamical systems %K Rauzy fractals %K Interval exchange transformations %K Peano curves %Z 37B10, 37E05 %Z Mathematics [math]/Dynamical Systems [math.DS]Journal articles %X We consider a substitution associated with the Arnoux-Yoccoz interval exchange transformation (IET) related to the tribonacci substitution. We construct the so-called stepped lines associated with the fixed points of the substitution in the abelianization (symbolic) space. We analyze various projections of the stepped line, recovering the Rauzy fractal, a Peano curve related to work in [Arnoux 88], another Peano curve related to the work of [McMullen 09] and [Lowenstein et al. 07], and also the interval exchange transformation itself. %G English %2 https://hal.science/hal-00965169/document %2 https://hal.science/hal-00965169/file/RPrevised25Juillet2009.pdf %L hal-00965169 %U https://hal.science/hal-00965169 %~ UNIV-TLSE2 %~ UNIV-TLSE3 %~ CNRS %~ INRIA %~ UNIV-AMU %~ INSA-TOULOUSE %~ INPL %~ IECN %~ INSMI %~ IMT %~ IML %~ I2M %~ UT1-CAPITOLE %~ UNIV-LORRAINE %~ TDS-MACS %~ INRIA-AUT %~ INSA-GROUPE %~ GDMM %~ UNIV-UT3 %~ UT3-INP %~ UT3-TOULOUSEINP