Restricted Roth type interval exchange maps from almost every direction on a fixed translation surface

For an interval exchange map (i.e.m.) of restricted Roth type — the explicit Diophantine condition of Marmi–Moussa–Yoccoz combining sub-polynomial growth of the Zorich continued-fraction cocycle with hyperbolicity (nonzero Lyapunov exponents) of the Kontsevich–Zorich cocycle — Marmi and Yoccoz (2016) proved that the cohomological equation admits Hölder solutions with essentially one derivative of loss, after a finite-dimensional correction. The class has full measure in the space of i.e.m.s, but the authors state that they do not know whether it is realized fiberwise: whether, on a given translation surface, the first-return maps of the linear flow are of restricted Roth type for almost every direction. Chaika–Eskin (2015) and Marmi–Ulcigrai–Yoccoz (2020) established this almost-every-direction property only for a strictly weaker condition (restricted absolute Roth type), which nevertheless suffices for the regularity conclusions when the Kontsevich–Zorich cocycle on the orbit closure is hyperbolic. The open problem is the original fibered question on this KZ-hyperbolic class: prove or refute that almost every direction on a fixed KZ-hyperbolic translation surface yields an i.e.m. of restricted Roth type in the original, relative-homology sense.

Background

An interval exchange map (i.e.m.) is an orientation-preserving, Lebesgue-measure-preserving bijection of a bounded interval that acts as a translation on each of $d$ open subintervals; for $d=2$ it is a circle rotation. i.e.m.s arise as first-return (Poincaré) maps of linear flows on translation surfaces — compact surfaces obtained by gluing polygons edge-to-edge by translations, carrying a flat metric with conical singularities at a finite set $\Sigma$ of marked points; equivalently a pair $(M,\zeta)$ of a compact Riemann surface $M$ with a holomorphic $1$-form $\zeta$ whose flat metric $|\zeta|$ is conical exactly at the zeros of $\zeta$ and the marked points $\Sigma$. The first-return map of the flow in a given direction to a transverse segment in good position — an interval, typically joining two singularities, on which every orbit returns and the induced map is a well-defined i.e.m. — exchanges $d = 2g+s-1$ intervals, where $g$ is the genus and $s=|\Sigma|$. The cohomological equation for a map $T$ asks for a solution $\psi$ of $\varphi = \psi\circ T - \psi$; it is a small-divisor problem, and for circle rotations its solvability in low regularity requires an arithmetic condition on the rotation number. An irrational number is of Roth type when it is not approximable by rationals to any exponent $2+\varepsilon$, $\varepsilon>0$; every irrational algebraic number and Lebesgue-almost every irrational has this property, and for rotations of Roth type the cohomological equation is solvable with essentially one derivative of loss.

Rauzy and Veech generalized the continued-fraction algorithm to i.e.m.s: inducing $T$ on a suitable subinterval produces a renormalization dynamics on the combinatorial and length data, and Zorich's acceleration of it has an explicit ergodic invariant measure. Iterating the induction encodes the i.e.m. into a sequence of nonnegative integer matrices $B(m,n)$ — the Zorich cocycle, the extended Kontsevich–Zorich (KZ) cocycle — acting on the space $\Gamma(T)\cong\mathbb{R}^d$ of piecewise-constant functions, naturally identified with the relative homology $H_1(M,\Sigma;\mathbb{R})$ of any suspending surface $M$; the invariant subspace $\mathrm{Im}\,\Omega(\pi)\cong H_1(M;\mathbb{R})$ (absolute homology, dimension $2g$; $\Omega(\pi)$ is the skew-symmetric matrix determined by the combinatorial data $\pi$) carries the classical KZ cocycle. By Oseledets' multiplicative ergodic theorem the cocycle has Lyapunov exponents, and Forni proved that for almost every i.e.m. the restricted cocycle on absolute homology is hyperbolic (all exponents nonzero).

Marmi–Moussa–Yoccoz (2005) introduced the Roth type condition for i.e.m.s — the higher-genus analogue of Roth type numbers — requiring: (a) sub-polynomial growth of the Zorich cocycle blocks, $\|B(n_k,n_{k+1})\| = O(\|B(0,n_k)\|^{\tau})$ for every $\tau>0$, along the sequence of times $(n_k)$ at which the blocks $B(n_k,n_{k+1})$ first have all entries strictly positive; (b) a spectral-gap contraction $\|B(0,n)|_{\Gamma^0}\| = O(\|B(0,n)\|^{1-\theta})$ for some $\theta>0$ on the hyperplane $\Gamma^0$ of mean-zero piecewise-constant functions (which forces unique ergodicity); and (c) for every $\tau>0$, the sub-polynomial bounds $\|B^s(m,n)\| = O(\|B(0,n)\|^{\tau})$ and $\|(\bar B(m,n))^{-1}\| = O(\|B(0,n)\|^{\tau})$ on the action induced by $B(m,n)$ on the stable subspace $\Gamma^s(T)$ (the vectors exponentially contracted by the cocycle) and on the inverse of its induced action on the quotient $\Gamma(T)/\Gamma^s(T)$. They proved the condition has full measure among i.e.m.s with fixed combinatorial data, and that for such $T$ the cohomological equation is solvable — after correction by a piecewise-constant function killing finitely many obstructions — with a loss of about two derivatives. The strengthened restricted Roth type condition adds (d): $\dim\Gamma^s(T) = g$, i.e. hyperbolicity of the cocycle (nonvanishing Lyapunov exponents); it was introduced for the linearization theory of generalized i.e.m.s (Marmi–Moussa–Yoccoz 2012) and also has full measure (Marmi–Yoccoz 2016, Remark 3.4). Marmi–Yoccoz (2016) proved that for a restricted Roth type i.e.m. and data $\varphi$ of class $C^r$, $r>1$, lying in the kernel of the boundary operator (a finite-codimension space eliminating the obstruction functionals), the solution is Hölder: $\varphi = \chi + \psi\circ T - \psi$ with $\chi$ piecewise constant and $\psi$ Hölder of some exponent $\bar\delta < r-1$ — a loss of essentially one derivative, close to the optimal circle-rotation result.

These full-measure statements are parametric: they hold for Lebesgue-almost every i.e.m. with any fixed combinatorial data, equivalently for almost every surface with respect to the natural invariant measure on a stratum of the moduli space — the locus of translation surfaces with a fixed number and orders of conical singularities. The source question is instead fibered: it fixes one surface and varies only the direction of the flow. The relevant object is then the $\mathrm{SL}(2,\mathbb{R})$-orbit closure of the surface: the group $\mathrm{SL}(2,\mathbb{R})$ acts on the moduli space by post-composing the flat charts with linear maps, every orbit closure carries a canonical invariant affine measure, and the Lyapunov exponents of the KZ cocycle with respect to this measure are well defined for the closure, with nonnegative part $\lambda_1\ge\dots\ge\lambda_g\ge 0$. Forni (1997) had earlier solved the cohomological equation for almost every direction on every given translation surface, in the Sobolev scale with a loss of at most $3+\varepsilon$ derivatives, later refined to $1+\varepsilon$ for almost every surface and direction (Forni 2021), but by methods that produce no explicit Diophantine condition on the induced i.e.m. Whether the explicit restricted Roth type condition holds fiberwise — for the first-return maps of almost every direction on a fixed surface — is exactly the gap Marmi–Yoccoz left open when comparing their theorem with Forni's, and is the subject of this problem.

Problem Statement

Prove or refute the fibered Diophantine statement that Marmi–Yoccoz (2016, §1.1) left open, on the class where it can hold.

Let $(M,\Sigma,\zeta)$ be a compact translation surface of genus $g\ge 2$ — a Riemann surface $M$ carrying a holomorphic $1$-form $\zeta$ that defines the flat structure, with $\Sigma$ the set of $s$ marked (singular) points — such that the Lyapunov exponents $\lambda_1\ge\dots\ge\lambda_g\ge 0$ of the Kontsevich–Zorich cocycle for the invariant measure on its $\mathrm{SL}(2,\mathbb{R})$-orbit closure satisfy $\lambda_g>0$ (no zero exponents; the KZ-hyperbolic case). For a direction $\theta$, let $T_\theta$ be the first-return map of the linear flow in direction $\theta$ to a segment in good position — a transverse interval on which the return map is a well-defined i.e.m., with $d=2g+s-1$ exchanged intervals. Is $T_\theta$ of restricted Roth type for Lebesgue-almost every $\theta$, i.e. does it satisfy all four conditions (a)–(d) of Marmi–Yoccoz (2016), §3.1, for the Zorich cocycle acting on relative homology $\Gamma(T_\theta)\cong H_1(M,\Sigma;\mathbb{R})$ — in particular condition (a): for every $\tau>0$, $\|B(n_k,n_{k+1})\| = O(\|B(0,n_k)\|^{\tau})$ along the positivity times $(n_k)$ of the cocycle, and not merely the weaker absolute condition (a)′ that published results establish?

Two documented qualifications delimit the question. First, the hypothesis $\lambda_g>0$ is necessary: on orbit closures with a zero exponent — e.g. the closed orbits (Teichmüller curves) of the square-tiled Eierlegende Wollmilchsau and Ornithorynque surfaces, finite covers of a flat torus on which all non-tautological Kontsevich–Zorich exponents vanish — condition (d) fails for almost every direction by Oseledets genericity (Chaika–Eskin), so the blanket form of the question without this hypothesis has a negative answer there; the genus-1 case is the classical one-dimensional theory. Second, the known positive results (Chaika–Eskin 2015; Marmi–Ulcigrai–Yoccoz 2020) already give, for a.e. direction on any KZ-hyperbolic surface, the weaker restricted absolute Roth type condition and with it all Hölder-regularity conclusions of the 2016 theorem; what remains open is precisely the original relative-homology condition — equivalently, whether the gap between condition (a) and its absolute weakening (a)′ can be closed fiberwise.

References

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