Eigenvalues and non-weak-mixing directions of non-arithmetic Veech surfaces: the Avila-Delecroix open problems

Avila and Delecroix (J. Amer. Math. Soc. 29 (2016), 1167-1208) proved that on every non-arithmetic Veech surface the translation flow is weakly mixing in almost every direction, that the exceptional set of directions has Hausdorff dimension strictly less than one, and that in every minimal direction the eigenvalue group is finitely generated with at most $[k:\mathbb{Q}]$ rationally independent eigenvalues, where $k$ is the trace field. Four questions raised or left open by that work and by subsequent literature — forming one coherent objective, a complete spectral and fractal description of the non-weak-mixing directions of non-arithmetic Veech surfaces and of the mechanism generating them — remain open: (1) the authors' expectation that every minimal direction that is not weakly mixing has exactly $[k:\mathbb{Q}]$ rationally independent eigenvalues, all continuous or all discontinuous, currently known only for linearly recurrent directions; (2) whether every non-arithmetic Veech group contains a Salem element, which would make the paper's positive-Hausdorff-dimension mechanism universal; (3) the exact Hausdorff dimension, or explicit bounds, of the exceptional set for specific surfaces such as the regular pentagon surface; and (4) the extent to which eigenvalues of general translation flows must arise from the strong-stable lattice of the Kontsevich-Zorich cocycle. No resolution of any component is known to date; the closest later works re-pose parts of (3) and (4) as open problems.

Background

A translation surface is a compact surface obtained by gluing finitely many Euclidean polygons along parallel edges by translations; equivalently, a pair $(S,\omega)$ of a compact Riemann surface $S$ with a nonzero holomorphic 1-form (Abelian differential) $\omega$. The straight-line or translation flow $\phi^\theta_\bullet$ is the flow along constant-slope vector fields in a chosen direction $\theta$ of the flat structure. Unfolding billiard motion in a rational polygon (all angles rational multiples of $\pi$) produces a translation surface, so billiards in regular polygons are special cases. A direction is minimal if it contains no saddle connection (no segment joining singularities in that direction); then every orbit avoiding the singularities is dense.

A real number $\nu$ is an eigenvalue of the directional flow if $f\circ\phi^\theta_t=e^{2\pi i\nu t}f$ almost everywhere for some nonzero $f\in L^2$; $f$ is an eigenfunction, and it may be continuous or genuinely discontinuous (measurable only). The flow is weakly mixing if the only eigenvalue is $0$, with multiplicity one — the strongest mixing-type property available here, since translation flows are never mixing (Katok). Eigenvalues form a countable subgroup of $\mathbb{R}$, whose rank over $\mathbb{Q}$ counts the rationally independent eigenvalues; the flow is topologically weakly mixing if there are no nonconstant continuous eigenfunctions, a strictly weaker property than weak mixing.

The group $SL(2,\mathbb{R})$ acts on translation surfaces by affinely deforming the polygons, and the stabilizer of $(S,\omega)$ is the Veech group, the image of the affine homeomorphism group in $SL(2,\mathbb{R})$. A Veech surface is one whose Veech group is a lattice (finite covolume); these are the most symmetric translation surfaces, and Veech proved for them a dichotomy: every direction is either completely periodic (the surface decomposes into cylinders) or uniquely ergodic. Veech surfaces are arithmetic when commensurable to $SL(2,\mathbb{Z})$ — essentially the square-tiled covers of the torus — and non-arithmetic otherwise; the surfaces $S_n$ obtained from billiards in the regular $n$-gon are non-arithmetic exactly for $n\ne 3,4,6$. The trace field $k$ of a non-arithmetic Veech surface — the field generated by the traces of its Veech group — satisfies $1\le[k:\mathbb{Q}]\le g$ (genus $g$) and coincides with the field of holonomy numbers of $\omega$. Möller proved that all Galois conjugates of the Veech group, other than the group itself, are non-discrete subgroups of $SL(2,\mathbb{R})$.

The analytic engine is renormalization. The Kontsevich-Zorich (KZ) cocycle records the evolution of the cohomology $H^1(S;\mathbb{R})$ along the Teichmüller geodesic flow on the $SL(2,\mathbb{R})$-orbit of the surface. The Veech criterion of Avila-Delecroix constrains eigenvalues: for a minimal direction, each eigenvalue $\nu$ is parametrized (via $\nu\,\mathrm{Im}(\omega)$) by a point of the weak-stable lamination of the KZ cocycle, modulo integer cohomology. For almost every surface in each stratum of the moduli space of Abelian differentials, Avila-Forni proved weak mixing in almost every direction by showing that this lamination avoids nonzero integer points — a probabilistic argument over parameter space that fails for a fixed Veech surface, where the orbit is a closed Teichmüller curve and the KZ cocycle degenerates (the non-twisting phenomenon).

Avila-Delecroix overcame this degeneracy for non-arithmetic Veech surfaces by constraining the locus of possible eigenvalues to the strong-stable lamination — the countable set of integer translates of the stable space of the cocycle — and by direct geometric estimates on that locus they proved: for every non-arithmetic Veech surface, the set of non-weak-mixing directions has Hausdorff dimension strictly less than one (so almost every direction is weakly mixing), and for quadratic trace fields — including all non-arithmetic Veech surfaces of genus 2 — the set of directions that are not even topologically weakly mixing has positive Hausdorff dimension. In any minimal direction the eigenvalue group is finitely generated, ratios of eigenvalues lie in $k$, and there are at most $[k:\mathbb{Q}]$ rationally independent eigenvalues; in the arithmetic case the eigenvalues come from a ramified cover of a torus and there are exactly two rationally independent continuous ones. For linearly recurrent directions — those with return times to any embedded horizontal interval $J$ bounded by $K/|J|$ for some constant $K$, equivalently those whose forward Teichmüller geodesic stays in a compact part of the moduli space — a criterion of Bressaud-Durand-Maass for linearly recurrent Cantor systems applies and gives a complete answer: either $0$ or exactly $[k:\mathbb{Q}]$ rationally independent eigenvalues, simultaneously continuous or simultaneously discontinuous.

The mechanism that produces many exceptional directions is arithmetic. A Salem number is a real algebraic integer $\lambda>1$ whose remaining conjugates have modulus at most one, at least one having modulus exactly one (equivalently, $\lambda$ is conjugate to $\lambda^{-1}$ and all other conjugates lie on the unit circle). A Salem element of a Veech group is a direct hyperbolic element — hyperbolic with positive dominant eigenvalue — all of whose Galois conjugates are elliptic, equivalently one whose dominant eigenvalue is a Salem number. Avila-Delecroix's Theorem 32 states that if the Veech group of a non-arithmetic Veech surface contains a Salem element, then both the set of directions admitting a continuous eigenfunction and the set of directions admitting a measurable discontinuous one have positive Hausdorff dimension. Salem elements exist for all non-arithmetic Veech surfaces with quadratic trace field (a result of Bressaud-Bufetov-Hubert, derived from Möller's non-discreteness), and computationally for the triangle groups $\Delta(2,q,\infty)$ with odd $q\le 15$ and $\Delta(q,\infty,\infty)$ with $q\le 15$, which covers the regular-polygon surfaces $S_n$ with $n\le 16$.

Later work sharpened the surrounding picture without touching the open cores. Avila-Leguil proved that for almost every interval exchange transformation and almost every translation surface in each stratum, the set of non-weak-mixing parameters has Hausdorff dimension strictly less than full — a statement about parameter-typical systems, not about a fixed Veech surface. Arana-Herrera-Chaika-Forni characterized translation surfaces that are weakly mixing in almost every direction: exactly those with no nonzero absolute integer cohomology class in the tautological plane spanned by the real and imaginary parts of $\omega$, settling a classical conjecture on rational billiards; their open-problems section re-poses the Hausdorff-dimension bound and the strong-stable question as open and beyond current methods. Athreya-Bédaride-Hooper-Hubert apply the Avila-Delecroix Salem mechanism to a new class of flat objects (translation prisms), taking the existence of a Salem element as a defining hypothesis; Forni's survey records that quantitative (polynomial) weak mixing is open in the rank-one case, which includes non-arithmetic Veech surfaces.

Problem Statement

Resolve the four open questions raised or left open by Avila-Delecroix's 'Weak mixing directions in non-arithmetic Veech surfaces' and by subsequent literature, concerning the exceptional — that is, not weakly mixing — directions of non-arithmetic Veech surfaces, formulated so that the four parts share one objective: a complete spectral and fractal description of those directions and of the mechanism generating them. The four parts are independent open questions: each can be resolved separately, and the shared objective is attained only when all four are resolved.

(1) Multiplicity/continuity dichotomy. Prove or disprove the authors' expectation: for every non-arithmetic Veech surface with trace field $k$ and every minimal direction in which the flow is not weakly mixing, there are exactly $[k:\mathbb{Q}]$ rationally independent eigenvalues, and the eigenfunctions are either all continuous or all genuinely discontinuous. The upper bound of $[k:\mathbb{Q}]$ is known in full generality (their Theorem 23); the matching lower bound and the regularity dichotomy are known only when the direction is linearly recurrent, equivalently when its forward Teichmüller geodesic is bounded in the moduli space (their Remark 7.1, via the Bressaud-Durand-Maass criterion).

(2) Salem universality. Determine whether every non-arithmetic Veech group contains a Salem element (a direct hyperbolic element, that is hyperbolic with positive dominant eigenvalue, all of whose Galois conjugates are elliptic; equivalently one whose dominant eigenvalue is a Salem number). A positive answer would make the positive-Hausdorff-dimension conclusion of their Theorem 32 — for which the existence of a Salem element is a sufficient hypothesis — applicable to all non-arithmetic Veech surfaces. Known for quadratic trace fields (including all non-arithmetic Veech surfaces of genus 2) and, computationally, for the triangle groups $\Delta(2,q,\infty)$ with odd $q\le 15$ and $\Delta(q,\infty,\infty)$ with $q\le 15$. Either a proof for trace fields of degree at least 3, or a fully certified counterexample (a non-arithmetic Veech group proved to contain no Salem element), resolves this component.

(3) Size of the exceptional set. Determine the Hausdorff dimension of the set of non-weak-mixing directions of a non-arithmetic Veech surface — exactly, or via explicit upper and lower bounds (concrete bounds, not merely the qualitative statement that the dimension is less than one) — for the standard families, that is, the regular polygon surfaces $S_n$ with $n\ne 3,4,6$, and in particular the pentagon surface $S_5$. This component is resolved by determining this dimension, exactly or via such explicit bounds, for the pentagon surface $S_5$; a determination for all $S_n$ with $n\ne 3,4,6$, or for a general non-arithmetic Veech surface, is a stronger result. This value is not determined by the Avila-Delecroix results and is not explicitly posed in their paper; known in general from it: the dimension is strictly less than one (their Theorem 2); under the Salem hypothesis (and for all quadratic trace fields, for the smaller set of directions that are not even topologically weakly mixing) it is positive (their Theorems 32 and 3). No explicit quantitative bound is known for any surface; the request for an explicit upper bound is re-posed as an open problem, flagged as beyond the reach of current methods, by Arana-Herrera-Chaika-Forni (their Open Problem 6.1), who note that it is unclear how to obtain an explicit upper bound for the set of directions with an eigenvalue coming from the strong-stable direction.

(4) Boundary of the strong-stable mechanism. Determine the reach of the strong-stable mechanism for eigenvalues of translation flows: the extent to which they must arise from the strong-stable lattice (the integer translates of the stable space of the Kontsevich-Zorich cocycle). Avila-Delecroix (section 1.3) expect that in the general case of translation flows one can construct examples of eigenvalues which do not come from the strong-stable lamination, and note that such examples already exist outside the recurrent regime (Guenais-Parreau; Bressaud-Durand-Maass). The concrete open question, posed by Arana-Herrera-Chaika-Forni (their Open Problem 6.1(6)), is to decide whether there exists a translation surface whose forward Teichmüller orbit is recurrent to a compact set, whose vertical flow is not weakly mixing, and which has an eigenvalue $\nu$ such that the parameter $\nu\,\mathrm{Im}(\omega)$ escapes every integer translate under the cocycle.

References

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