Minimal interval exchange transformations with flips

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.