Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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

share|improve this question

1 Answer 1

up vote 5 down vote accepted

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.

share|improve this answer
    
where can i find the proof of this fact? –  user8991 Mar 17 at 5:51
1  
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 at 21:47

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.