# Constructing an interval exchange given a prescribed trajectory

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

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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.
If $E$ is the set of endpoints of intervals, then $|E|=k-1$. $E\cup T^{-1}E\cup \ldots\cup T^{-(n-1)}E$ has cardinality at most $n(k-1)$, and so the complement in [0,1] has at most $1+(k-1)n$ intervals. If two points are not separated by one of these endpoints, they have the same $n$-step coding. –  Anthony Quas Mar 17 '14 at 21:47