Dynamical decomposition of generalized interval exchange transformations

TL;DR

This work extends Rauzy–Veech renormalization to generalized interval exchange transformations by introducing interval exchange transformations with gaps (g-GIETs) and a renormalization scheme that can isolate multiple dynamical components. It proves a decomposition theorem: any GIET on [0,1) splits into a domain of transition and finitely many domains of recurrence, each carrying either a unique quasiminimal or a periodic behavior, with a corresponding tower representation that clarifies the internal dynamics. The results generalize classical foliation decompositions (Levitt, Gardiner, Gutierrez) to the Poincaré maps level and provide explicit bounds on the number of periodic domains and non-atomic ergodic measures, while connecting to Poincaré–Yoccoz-type semi-conjugacy theorems for RV-stable g-GIETs. The approach yields a constructive renormalization framework that identifies recurrent orbit closures via the leftmost endpoint of renormalization limits and yields a structured, finite decomposition of the domain into dynamically coherent pieces. Overall, the paper deepens the understanding of GIET dynamics beyond infinite completeness, offering new tools for analyzing flows on surfaces through combinatorial renormalization and tower representations.

Abstract

We develop a renormalization scheme which extends the classical Rauzy-Veech induction used to study interval exchange tranformations (IETs) and allows to study generalized interval exchange transformations (GIETs) $T: [0,1) \to [0,1)$ with possibly more than one quasiminimal component (i.e. not infinite-complete, or, equivalently, not semi-conjugated to a minimal IET). The renormalization is defined for more general maps that we call interval exchange transformations with gaps (g-GIETs), namely partially defined GIETs which appear naturally as the first return map of $C^r$-flows on two-dimensional manifolds to any transversal segment. We exploit this renormalization scheme to find a decomposition of $[0,1)$ into finite unions of intervals which either contain no recurrent orbits, or contain only recurrent orbits which are closed, or contain a unique quasiminimal. This provides an alternative approach to the decomposition results for foliations and flows on surfaces by Levitt, Gutierrez and Gardiner from the $1980s$.

Figures (22)

• Figure 1: A g-GIET on $4$ intervals. The top line represents the intervals $I^t$ (and the $3$ gaps), the bottom line the intervals $I^b$ (also with $3$ gaps). Intervals with the same color are mapped to each other via an orientation preserving diffeomorphism.
• Figure 2: Three examples of flows on the torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$: an irrational linear flow (left), a Cherry flow (center) and a Denjoy flow (right).
• Figure 3: Denjoy-like and Cherry-like flows in $g=2$ (reproduced from Carrand, courtesy of J. Carrand).
• Figure 4: The two chamber surface (polygonal representation). Consider the grey region in the center bounded by two parallel lines in direction $\theta$. For directions $\theta \in S^1$ chosen in a carefully constructed exceptional set (uncountable, but of measure zero) the union of forward and backward iterates of this region with respect to $f^t_{\theta}$, where one iteration is depicted on the right, form an infinite band which winds around the surface. Its complement consists of two Cantor sets $\Omega^+$ and $\Omega^-$, each contained in one of the chambers.
• Figure 5: Topological representation of the two chamber surface. The pink curve represents its dynamical decomposition (see also Theorem \ref{['thm:gardiner']}).

Theorems & Definitions (73)

• Theorem 1.1: Decomposition theorem
• Theorem 1.2: Structure Theorem
• Theorem 1.3
• Corollary 1.4
• Corollary 1.5
• Definition 2.1
• Theorem 2.2: Gardiner, gardiner
• Definition 3.1
• Definition 3.2
• Definition 3.3