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Description
Interval Translations Maps (ITMs) are a natural generalisation of the well-known Interval Exchange Transformations (IETs). They are obtained by dropping the bijectivity assumption for IETs. As such, they are exactly the finite piecewise isometries of the interval. There are two types of ITMs, 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 set of finite-type maps contains an open and dense subset of the parameter space of ITMs with a fixed number of intervals, which resolves in positive the topological version of a long-standing conjecture due to Boshernitzan and Kornfeld.
This is a joint work with Kostiantyn Drach and Sebastian van Strien.