In this paper we characterize manifolds (topological or smooth, compact or not, with or without boundary) which admit flows having a dense orbit (such manifolds and flows are called transitive) thus fully answering some questions by Smith and Thomas. Namely, it is shown that a surface admits a transitive flow (which can be got smooth) if and only if it is connected and it is neither homeomorphic to the sphere nor the projective plane nor embeddable in the Klein bottle (or, alternatively, if it is connected and includes two orientable topological circles intersecting transversally at exactly one point). We also prove that any (connected) manifold with dimension at least 3 admits a transitive flow, which can be got smooth if the manifold admits a smooth structure. In particular, this allows us to characterize \omega -limit sets with nonempty interior for flows in a given n -manifold (as they can be described by the property of being the closure of its transitive n -submanifolds).

We show that where C is a positive constant and , and where k is an odd natural number, are the unique -cycles having the form where f _j are continuous selfmaps of the interval , .

Oriented interval exchange transformations (i.e.t.s) admitting infinite orbit of the Rauzy–Veech operator are minimal. In this paper, we analyse what happens for non-oriented i.e.t.s. In particular, we prove an analogous result for i.e.t.s with flips (Theorem A). We also show that the behaviour of i.e.t.s with flips is different to the one of oriented i.e.t.s, see Proposition 11.

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 Gutierrezet 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.