A priori bounds for GIETs, affine shadows and rigidity of foliations in genus 2

TL;DR

This paper establishes rigidity results for genus two foliations and GIETs, showing that under certain conditions, these systems are conjugate to simpler models, and reduces the conjecture on GIET rigidity to affine IETs analysis.

Contribution

It generalizes Herman's theorem to genus two, proves a conjecture on GIET rigidity, and links GIET behavior to affine IETs, using renormalization techniques.

Findings

Minimal genus two foliations are conjugate to linear foliations under full measure conditions.

GIETs topologically conjugate to certain IETs are $C^1$-conjugate under an obstruction.

Dichotomy in GIET renormalization: recurrence or approximation by affine IETs.

Abstract

We prove a rigidity result for foliations on surfaces of genus two, which can be seen as a generalization to higher genus of Herman's theorem on circle diffeomorphisms and, correspondingly, flows on the torus. We prove in particular that, if a smooth, orientable foliation with non-degenerate (Morse) singularities on a closed surface of genus two is minimal, then, under a full measure condition for the rotation number, it is differentiably conjugate to a linear foliation. The corresponding result at the level of Poincar\'e sections is that, for a full measure set of interval exchange transformations with 4 or 5 continuity intervals and irreducible combinatorics, any generalized interval exchange transformation which is topologically conjugate to a IET from this set and satisfies an obstruction given by a boundary operator is $\mathcal{C}^1$-conjugate to it. This in particular settles a conjecture by Marmi, Moussa and Yoccoz in genus two. Our results also show that this conjecture on the rigidity of GIETs can be reduced to the study of affine IETs, or more precisely of Birkhoff sums of piecewise constant observables over standard IETs, in genus $g \geq 3$. Our approach is via renormalization, namely we exploit a suitable Oseledets regular acceleration of the Rauzy-Veech induction on the space of GIETs. For infinitely renormalizable, irrational GIETs of any number of intervals $d\geq 2$ we prove a dynamical dichotomy on the behaviour of the orbits under renormalization, by proving that either an orbit is recurrent to certain bounded sets in the space of GIETs, or it diverges and it is approximated (up to lower order terms) by the orbit of an affine IET (a case that we refer to as affine shadowing).