Eigenvalue multiplicity and continuity dichotomy for minimal non-weak-mixing directions on non-arithmetic Veech surfaces (Avila–Delecroix conjecture)

For a non-arithmetic Veech surface with holonomy field $k$, Avila and Delecroix (JAMS 2016) predicted that along any minimal direction in which the directional flow is not weakly mixing, the eigenvalue group contains exactly $[k:\mathbb{Q}]$ eigenvalues linearly independent over $\mathbb{Q}$, and that either every nonzero eigenvalue admits a continuous eigenfunction or none does. The upper bound $[k:\mathbb{Q}]$ on the number of independent eigenvalues, and the full statement under a linear-recurrence (bounded forward Teichmüller geodesic) hypothesis, are theorems from the same paper; the general statement has remained open. The problem is to prove the conjecture for all non-arithmetic Veech surfaces and all minimal non-weak-mixing directions, or to refute it by an explicit counterexample.

Background

A translation surface is a closed surface obtained from finitely many polygons in the complex plane by gluing pairs of parallel sides through translations; equivalently, it is a compact Riemann surface together with a nonzero holomorphic 1-form $\omega$ whose zeros are the singularities. The straight-line or directional flow $\varphi_t^\theta$ moves every non-singular point at unit speed in the direction $\theta$; its first-return map to a transverse segment is an interval exchange transformation. A real number $\nu$ is an eigenvalue of the directional flow if there exists a nonzero measurable function $f$ with $f(\varphi_t^\theta x)=e^{2\pi i\nu t}f(x)$ for almost every $x$ and all $t$; $\nu$ is a continuous eigenvalue if $f$ can additionally be chosen continuous. On a compact surface the eigenvalues form a subgroup of $\mathbb{R}$, and the corresponding factor is the Kronecker factor of the flow, that is, its maximal measurable almost-periodic factor (a translation on a compact abelian group). The flow is weakly mixing exactly when it has no nonzero eigenvalues, so the eigenvalue group measures the failure of weak mixing.

The Veech group of a translation surface is the set of derivatives of its affine self-maps (homeomorphisms that become translations in the polygon charts); it is a discrete subgroup of $\mathrm{SL}(2,\mathbb{R})$. A Veech surface (or lattice surface) is one whose Veech group is a lattice; the surfaces obtained by unfolding billiards in regular polygons are examples. By the Veech dichotomy, in every direction a Veech surface either decomposes into cylinders of closed trajectories (a completely periodic direction) or the directional flow is minimal (all non-singular orbits dense) and uniquely ergodic (only one invariant probability measure). Attached to each Veech surface is its holonomy field $k$ (also called the trace field): the number field generated by the traces of the derivatives of the affine self-maps, equivalently the smallest subfield of $\mathbb{R}$ such that an affine image of the surface decomposes into polygons with edge vectors in $k^2$. The surface is arithmetic, meaning affinely equivalent to a square-tiled surface (a branched cover of a flat torus), exactly when $k=\mathbb{Q}$, and non-arithmetic when $k\neq\mathbb{Q}$; for instance the billiard in the regular pentagon unfolds to a non-arithmetic Veech surface with $[k:\mathbb{Q}]=2$.

For typical (almost every) surfaces and directions, weak mixing was proved by Avila and Forni. For the rigid class of Veech surfaces, Avila and Delecroix proved that on every non-arithmetic Veech surface almost every direction is weakly mixing and that the exceptional set of non-weak-mixing directions has Hausdorff dimension (a fractal dimension) strictly smaller than one; conversely, whenever the Veech group contains a Salem element (a hyperbolic element whose derivative has a Salem number as expanding eigenvalue, a Salem number being a real algebraic integer greater than 1 all of whose other conjugates have modulus at most 1 with at least one of modulus exactly 1), the set of non-weak-mixing directions has positive Hausdorff dimension, and among these exceptional directions both directions carrying a continuous eigenfunction and directions admitting only measurable, discontinuous eigenfunctions occur. For the exceptional directions they proved structural restrictions (Theorem 23 of that paper): in every minimal direction the eigenvalue group is finitely generated; for a non-arithmetic Veech surface the ratio of any two nonzero eigenvalues lies in $k$, so there are at most $[k:\mathbb{Q}]$ rationally independent eigenvalues; for arithmetic surfaces all eigenfunctions are lifts from the maximal torus cover and there are exactly two rationally independent continuous eigenvalues. The mechanism is renormalization: under the Teichmüller geodesic flow (the diagonal part of the $\mathrm{SL}(2,\mathbb{R})$-action on the moduli space of translation surfaces), eigenvalues must correspond to integer vectors in the lattice $W_{\mathbb{Z}}=W\cap H^1(S;\mathbb{Z})$, where $W=\bigoplus_\sigma V^\sigma$ is the direct sum of the tautological plane $V=\mathbb{R}\,\mathrm{Re}(\omega)\oplus\mathbb{R}\,\mathrm{Im}(\omega)$ and its Galois conjugate planes, coupled by the Kontsevich–Zorich cocycle (the linear cocycle on cohomology describing this renormalization). Under the additional hypothesis of linear recurrence — equivalently, boundedness of the forward Teichmüller geodesic in moduli space — the authors proved the exact expected picture (Remark 7.1 of that paper): either the flow is weakly mixing or there are exactly $[k:\mathbb{Q}]$ rationally independent eigenvalues, and these are simultaneously all continuous or all discontinuous.

What remains unknown is whether this exact multiplicity and the continuity dichotomy hold for every minimal non-weak-mixing direction of every non-arithmetic Veech surface, without any recurrence assumption. The conjecture was stated by Avila and Delecroix in the introduction of their 2016 paper as an expectation.

Problem Statement

Let $(S,\omega)$ be a non-arithmetic Veech surface, that is, a translation surface whose Veech group is a lattice in $\mathrm{SL}(2,\mathbb{R})$ and whose holonomy field $k$ is a proper extension of $\mathbb{Q}$. Let $\theta$ be a minimal direction in which the directional flow $\varphi_t^\theta$ is not weakly mixing; recall that $\nu\in\mathbb{R}$ is an eigenvalue if some nonzero measurable $f$ satisfies $f(\varphi_t^\theta x)=e^{2\pi i\nu t}f(x)$ almost everywhere, and a continuous eigenvalue if $f$ can be chosen continuous. Prove or disprove the conjecture of Avila and Delecroix:

(i) the eigenvalue group of $\varphi_t^\theta$ contains exactly $[k:\mathbb{Q}]$ eigenvalues that are linearly independent over $\mathbb{Q}$ (equivalently, the $\mathbb{Q}$-vector space spanned by the eigenvalue group has dimension $[k:\mathbb{Q}]$); and

(ii) either every nonzero eigenvalue of $\varphi_t^\theta$ is continuous, or every nonzero eigenvalue is discontinuous (no nonzero eigenvalue admits a continuous eigenfunction).

The statement concerns the class of all non-arithmetic Veech surfaces and all minimal non-weak-mixing directions. A result covering only directions with bounded forward Teichmüller geodesic (the linearly recurrent case), only quadratic trace fields, or only particular surfaces such as those of regular-polygon billiards settles only the already-known partial cases and does not resolve the problem; a valid refutation is an explicit surface and direction violating clause (i) or clause (ii).

References

1. Artur Avila and Vincent Delecroix, Weak mixing directions in non-arithmetic Veech surfaces, Journal of the American Mathematical Society 29 (2016), no. 4, 1167–1208. DOI: 10.1090/jams/856, https://doi.org/10.1090/jams/856 (preprint arXiv:1304.3318, https://arxiv.org/abs/1304.3318)
2. Artur Avila and Giovanni Forni, Weak mixing for interval exchange transformations and translation flows, Annals of Mathematics (2) 165 (2007), 637–664. DOI: 10.4007/annals.2007.165.637, https://doi.org/10.4007/annals.2007.165.637
3. W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Inventiones Mathematicae 97 (1989), 553–583. DOI: 10.1007/BF01388890, https://doi.org/10.1007/BF01388890
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