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Wandering intervals in affine extensions of self-similar interval exchange maps: the cubic Arnoux–Yoccoz map

Published online by Cambridge University Press:  08 May 2017

MILTON COBO ,
RODOLFO GUTIÉRREZ-ROMO  and
ALEJANDRO MAASS
MILTON COBO
Affiliation:
Departamento de Matemática, Universidade Federal do Espírito Santo, Av. Fernando Ferrari 514, Goiabeiras, Vitória, Brasil email milton.e.cobo@gmail.com
RODOLFO GUTIÉRREZ-ROMO
Affiliation:
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, CNRS-UMI 2807, Universidad de Chile, Beauchef 851, Santiago, Chile email rgutierrez@dim.uchile.cl, amaass@dim.uchile.cl
ALEJANDRO MAASS
Affiliation:
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, CNRS-UMI 2807, Universidad de Chile, Beauchef 851, Santiago, Chile email rgutierrez@dim.uchile.cl, amaass@dim.uchile.cl
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Abstract

In this article, we provide sufficient conditions on a self-similar interval exchange map, whose renormalization matrix has complex eigenvalues of modulus greater than one, for the existence of affine interval exchange maps with wandering intervals that are semi-conjugate with it. These conditions are based on the algebraic properties of the complex eigenvalues and the complex fractals built from the natural substitution emerging from self-similarity. We show that the cubic Arnoux–Yoccoz interval exchange map satisfies these conditions.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 7 , October 2018 , pp. 2537 - 2570
Copyright
© Cambridge University Press, 2017 

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