Wandering intervals in affine extensions of self-similar interval exchange maps: the cubic Arnoux-Yoccoz map. (English) Zbl 1402.37052
In this paper the authors find sufficient conditions for a self-similar interval exchange map (i.e.m.) to have an affine i.e.m. topological extension with wandering intervals. This type of problem has been studied by several researchers (see for instance [R. Camelier and C. Gutierrez, Ergodic Theory Dyn. Syst. 17, No. 6, 1315–1338 (1997; Zbl 0895.58019); M. Cobo, Ergodic Theory Dyn. Syst. 22, No. 2, 375–407 (2002; Zbl 1136.37304); X. Bressaud et al., Ergodic Theory Dyn. Syst. 30, No. 3, 665–686 (2010; Zbl 1200.37002)]). To achieve their results, the authors implement the strategy of Camelier and Gutiérrez [loc. cit.] and Bressaud et al. [loc. cit.] and use the geometric models for substitutions of P. Arnoux et al. [Exp. Math. 20, No. 1, 97–120 (2011; Zbl 1266.37008)]. Lastly, they apply their results to the cubic Arnoux-Yoccoz i.e.m. [P. Arnoux and J.-C. Yoccoz, C. R. Acad. Sci., Paris, Sér. I 292, 75–78 (1981; Zbl 0478.58023)].
Reviewer: Steve Pederson (Atlanta)
MSC:
| 37E05 | Dynamical systems involving maps of the interval |
| 28A80 | Fractals |
| 37E20 | Universality and renormalization of dynamical systems |
References:
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