Denis Volk: Interval translation maps of three intervals

Interval translation maps (ITMs) are non-invertible generalizations of interval exchanges. There are two classes of them, `finite type' and `infinite type'. The dynamics of `finite type' ITMs is similar to interval exchanges. Infinite type exhibit some non-trivial attractors with substitution shift dynamics on them. The finiteness conjecture says that almost every ITM is finite type. I intend to give an introduction to the field, discuss the proof of the conjecture for three intervals, and outline the directions of further research.