A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces

Abstract

ABSTRACT. We present a Galois-theoretical criterion for the simplicity of the Lyapunov spectrum of the Kontsevich-Zorich cocycle over the Teichmüller flow on the SL 2 (R)-orbit of a square-tiled surface. The simplicity of the Lyapunov spectrum has been proved by A. Avila and M.Viana with respect to the so-called Masur-Veech measures associated to connected components of moduli spaces of translation surfaces, but is not always true for square-tiled surfaces of genus 3. We apply our criterion to square-tiled surfaces of genus 3 with one single zero. Conditionally to a conjecture of Delecroix and Lelièvre, we prove with the aid of Siegel's theorem (on integral points on algebraic curves of genus > 0) that all but finitely many such square-tiled surfaces have simple Lyapunov spectrum.

Diffusion for chaotic plane sections of 3-periodic surfaces

“…Now we apply the Galois-theoretical criterium of the simplicity of Lyapunov spectra from [MaMöYo15] (Theorem 2.17). This criterium develops the idea suggested in [AvVi07].…”

“…We have to provide first the Galois-pinching matrix (see Definition 2.12 from [MaMöYo15]) for the cocycle B. In accordance with [MaMöYo15], the matrix of the cocycle is Galois-pinching if its characteristic polynomial is irreducible over Q, has only real roots, and its Galois group is largest possible (see Chapter 4.1 and in particular Definition 4.1 in [MaMöYo15]). We work with the cocycle without orientation because Lemma 28 implies that all the properties of spectrum of the cocycle A are the same.…”

“…• The Galois group is isomorphic to S 3 since ∆ 1 = 1940 is not a square of any rational number. Now we need a matrix B 2 that is twisting with respect to B 1 (see Chapter 4.2 in [MaMöYo15]. especially Theorem 4.6).…”

The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces

“…Indeed, G is contained in this stabilizer. As R-Lie groups, the stabilizer of e in SL(2, C) has only three types of Zariski closed subgroups: the two trivial subgroups 6 , and (given any vector f independent of e) the subgroup G f of the stabilizer formed of elements g such that g.f − f is a real multiple of e (there is a one-parameter family, parametrized by the 1-dimensional real projective space, of such subgroups). Here, the existence of a parabolic element guarantees that G is not reduced to the identity.…”

An origami of genus 3 with arithmetic Kontsevich–Zorich monodromy

“…Indeed, these facts follow from the analysis of the discriminants ∆ 1 (χ p1 ) = (−11) 2 − 4 × (29 − 2) = 13, ∆ 1 (χ p2 ) = (−2) 2 − 4 × (−16 − 2) = 2 2 × 19 and ∆ 2 (χ p1 ) = (29+2) 2 −4×(−11) 2 = 3 2 ×53, ∆ 2 (χ p2 ) = (−16+2) 2 −4×(−2) 2 = 6 2 ×5 (cf. [MMY,§6.7]). By the Zariski density criterion of Prasad-Rapinchuk [PR,Theorem 9.10] (see also [Ri,Theorem 1.5]), we have that ρ(Aff(O 1 )) is Zariski-dense in Sp(H (0) 1 (O 1 , R)).…”