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My work is mostly about extending the renormalization theory of interval exchange transformations to settings where the classical assumptions break down. I study generalized, infinite-type and non-orientable interval exchanges and, together with my collaborators, develop new versions of Rauzy–Veech induction for them. These systems are interesting not only as generalizations of IETs or in terms of first return maps of flows on surfaces, but also as simple models for broader questions about recurrence, ergodicity and renormalization in dynamical systems.
More on IETs: An interval exchange transformation (IET) is a simple dynamical system obtained by cutting an interval into finitely many...
...subintervals and rearranging them by translations. As depicted below, IETs arise naturally as first-return maps of measure-preserving flows on surfaces. Despite their elementary definition, they display very rich dynamical behaviour and form one of the simplest settings in which phenomena such as minimality and unique ergodicity can be studied explicitly.
The main tool for studying IETs is Rauzy–Veech induction, a renormalization procedure that replaces an IET by its first-return map to a smaller subinterval. Iterating this construction encodes the long-term dynamics of the transformation through a sequence of combinatorial operations and associated matrix products (see also page 27 onwards in the slides under Talks). A foundational result, proved independently by H. Masur and W. Veech in 1982, states that almost every irreducible IET is uniquely ergodic. A key idea in the proof is to pass to one dimension higher up, relating Rauzy–Veech induction on the space of IETs to the Teichmüller geodesic flow on moduli spaces of so-called translation surfaces. This close connection with Teichmüller dynamics has made classical IETs one of the most successful models at the interface of dynamics, geometry, and combinatorics.
For further reading on IETs and translation surfaces, I recommend Yoccoz, Interval Exchange Maps and Translation Surfaces, and Zorich, Flat Surfaces.
Articles and Theses
(C. Fougeron, S. Schmidhuber, 2026) Rauzy-Veech Induction for Infinite-type IETs
We study interval exchange transformations with infinitely many exchanged intervals. These systems form a very broad class: by a famous theorem of Arnoux, Ornstein and Weiss, every invertible aperiodic measure-preserving transformation is measurably isomorphic to an infinite-type IET. However, many of the tools from the classical finite-dimensional theory are no longer available. In this article, we extend Rauzy–Veech induction to the infinite setting and use it to study basic dynamical questions such as minimality and unique ergodicity. In particular, we prove an analogue of Keane's minimality criterion and, for a natural family of infinite-type IETs, show that unique ergodicity is topologically generic.
(C. Fougeron, S. Schmidhuber, C. Ulcigrai, 2025, submitted) Dynamical Decomposition of Generalized Interval Exchange Transformations
We study generalized interval exchange transformations whose dynamics can split into several quasiminimal components. The usual Rauzy–Veech induction is not well suited to this situation since it may "get stuck" at a quasiminimal component before exploring the entire space. Hence we introduce a renormalization procedure which successively removes these quasiminimal regions, producing maps which we call GIETs with gaps. This in turn allows us to classify the different possible recurrent dynamics and to decompose the interval regions with different recurrent behaviour. In particular, the method gives a combinatorial way to recover decomposition results for flows and foliations on surfaces by Levitt, Gutierrez and Gardiner from the 1980s.
(S. Schmidhuber, Master's Thesis, 2023) On the Structure of Foliations on Dilation Surfaces
A dilation surface is obtained by gluing polygons in the plane by translations and rescaling of lengths, so it carries a natural affine structure. 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.
Work in Progress
(M. Jay, O. Paris-Romaskevich, S.Schmidhuber, Y. Tumarkin, work in progress) Suspending Interval Exchange Transformations with Flips
We study interval exchange transformations with flips, which are IETs where we allow the orientation of the intervals to be reversed. Their dynamics is very different to the orientable case: A.Nogueira proved in 1989, using a natural extension of Rauzy-Veech induction, that generically these maps are not minimal. Since however in the non-minimal cases the dynamics can be decomposed into smaller minimal parts, to gain a full understanding of the dynamics one needs to understand the cases which are minimal. We aim to find specific criterias for when a given permutation admits minimal length parameters, using a geometrical approach via constructing suspension surfaces for flip IETs, together with a combinatorial analysis of their Rauzy graphs.
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