Minimal interval exchange transformations with flips

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.

Existence of minimal flows on nonorientable surfaces

“…For the case of i.e.t. 's with flips, A. Linero and the third author have recently obtained [17] the following result, which will play an essential role in this paper. .…”