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Minimal interval exchange transformations with flips

Published online by Cambridge University Press:  03 April 2017

ANTONIO LINERO BAS  and
GABRIEL SOLER LÓPEZ Open the ORCID record for GABRIEL SOLER LÓPEZ [Opens in a new window]
ANTONIO LINERO BAS
Affiliation:
Departamento de Matemáticas, Universidad de Murcia (Campus de Espinardo), 30100-Espinardo-Murcia, Spain email lineroba@um.es
GABRIEL SOLER LÓPEZ
Affiliation:
Departamento de Matemática Aplicada y Estadística, Paseo Alfonso XIII, 52, 30203-Cartagena, Spain email gabriel.soler@upct.es
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Abstract

We consider interval exchange transformations of $n$ intervals with $k$ flips, or $(n,k)$-IETs for short, for positive integers $k,n$ with $k\leq n$. Our main result establishes the existence of minimal uniquely ergodic $(n,k)$-IETs when $n\geq 4$; moreover, these IETs are self-induced for $2\leq k\leq n-1$. This result extends the work on transitivity in Gutierrez et al [Transitive circle exchange transformations with flips. Discrete Contin. Dyn. Syst. 26(1) (2010), 251–263]. In order to achieve our objective we make a direct construction; in particular, we use the Rauzy induction to build a periodic Rauzy graph whose associated matrix has a positive power. Then we use a result in the Perron–Frobenius theory [Pullman, A geometric approach to the theory of non-negative matrices. Linear Algebra Appl. 4 (1971) 297–312] which allows us to ensure the existence of these minimal self-induced and uniquely ergodic $(n,k)$-IETs, $2\leq k\leq n-1$. We then find other permutations in the same Rauzy class generating minimal uniquely ergodic $(n,1)$- and $(n,n)$-IETs.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 8 , December 2018 , pp. 3101 - 3144
Copyright
© Cambridge University Press, 2017 

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