Research

We prove that unique ergodicity is topologically generic for IETs which reverse the order of their tail intervals (tail-reversing IETs). We generalize Rauzy-Veech to this setting and prove a minimality criterion as a generalization of Keane's criterion in the finite setting. We then define infinite-type Rauzy diagrams and obtain our genericity result through a combinatorial analysis of these diagrams. Moreover, we derive an explicit condition for a tail-reversing IET to be uniquely ergodic by studying the diameter of its induction matrices.

We extend Rauzy-Veech induction to GIETs with possible more than one quasiminimal component. The induction is defined for more general maps which we call gap-GIETs (partially defined GIETs). We then construct a renormalization scheme where we obtain gap-GIETs by successively removing intervals from GIETs, allowing us to find a decomposition of [0,1) into a finite union of intervals which either contain no recurrent orbits, or contain only recurrent orbits which are closed, or contain a unique quasiminimal. 

We prove a structure theorem for the directional foliations on dilation surfaces using a decomposition theorem established by C.J. Gardiner in the 1980s. We show that given a directional foliation on any dilation surface, there exists a decomposition of the surface into finitely many subsurfaces on which the foliation structure is in one of four possible cases: completely periodic, Morse-Smale, minimal or Cantor-like.