CSDC Seminar: Density of Stable Interval Translation Maps (Leon Staresinic)

Interval Translations Maps (ITM’s) are a natural generalisation of the well-known Interval Exchange Transformations (IET’s). They are obtained by dropping the bijectivity assumption for IET’s. As such they are exactly the finite piece-wise isometries of the interval.

There are two types of ITM’s, finite-type and infinite-type ones. They are classified by their non-wandering sets: it is a finite union of intervals for finite-type maps, and contains a Cantor set for infinite-type maps.

One of the basic questions in the field is: How prevalent is each type of map in the parameter space?

In this work, we show that the stable maps form a dense set in the parameter space of ITM’s with a fixed number of intervals, thus showing the prevalence of finite type maps in the topological sense. The key ingredient of this work is a theorem about linear independence of certain critical itinerary vectors.