Let $f$ be an interval exchange transformation of $[0,1]$. Is there an explicit formula giving $f^k(0)$ in function of $k$?
If not, are there particular cases where this formula is simple? (except when $f$ is a rotation).
I am mostly interested in the case when $f$ is periodic, and with the permutation ABC -> CBA where A,B,C are intervals of different lengths (with the interval B being translated).
For this particular interval exchange, we have: $f^k(0)=k_1 \lambda_1+k_2 \lambda_2 +k_3 \lambda_3$. where the $\lambda_i$ are the translation vector of $f$ on each of the three intervals. Is it possible to have an estimate the $k_i$?