Pedro Peres

Existence of non-trivial embeddings of Interval Exchange Transformations into Piecewise Isometries

Piecewise isometries (PWIs) are higher dimensional generalizations of one-dimensional interval exchange transformations (IETs). Unlike IETs, which are typically ergodic, it is conjectured that Lebesgue measure on the exceptional set is typically not ergodic in some families of PWIs - there can be non-smooth invariant curves that prevent trajectories from spreading across the whole of the exceptional set.
Recently, we proved that almost every IET, with an associated translation surface of genus g>1, can be non-trivially and isometrically embedded in a family of PWIs. In particular, this establishes the existence of invariant curves, which are not unions of circle arcs or line segments, for PWIs. In this talk, we outline the main steps of this proof.
(Joint work with Ana Rodrigues).