Given a prescribed trajectory, is it possible to construct an interval exchange having this trajectory?


For example, given a 3-letter word (like aaabbbccabcaaa ), is it possible to construct a 3- interval exchange with a point having this word as the beginning of its trajectory?


what necessary conditions on a given word to be the trarjectory of a IET can be found?


For the relation between coding and interval exchange, see e.g. : <http://combinat.sagemath.org/doc/reference/combinat/sage/combinat/iet/tutorial.html#orbit-and-symbolic-coding>



Constructing an interval exchange given a prescribed trajectory
The *complexity* of an infinite sequence $x$ is a sequence $C(n)$, where $C(n)$ is the number of distinct blocks of length $n$ in $x$. For an interval exchange with $k$ symbols, it's not hard to show that $C(n)=(k-1)n+1$. If your word has more complexity than this, it can never appear as the coding sequence of an IET.


