Moduli Space Of Translation Surfaces Research Articles

New phenomena in deviation of Birkhoff integrals for locally Hamiltonian flows

Abstract We consider smooth locally Hamiltonian flows on compact surfaces of genus g ≥ 1 {g\geq 1} to prove a deviation formula of Birkhoff integrals for smooth observables. Our work generalizes results of [G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2) 155 2002, 1, 1–103] and [A. I. Bufetov, Limit theorems for translation flows, Ann. of Math. (2) 179 2014, 2, 431–499] (and a recent result in [K. Frączek and C. Ulcigrai, On the asymptotic growth of Birkhoff integrals for locally Hamiltonian flows and ergodicity of their extensions, preprint 2021]) which prove the existence of the deviation spectrum of Birkhoff integrals for observables whose jets vanish at sufficiently high order around fixed points of the flow. They showed that ergodic integrals display a power deviation spectrum with exactly g positive exponents related to the positive Lyapunov exponents of the cocycle so-called Kontsevich–Zorich, a renormalization cocycle over the Teichmüller flow on a stratum of the moduli space of translation surfaces. Our paper extends the study of the deviation spectrum of ergodic integrals beyond the case of observables whose jets vanish at sufficiently high order around fixed points. We prove the existence of some extra terms in the deviation spectrum related to the non-vanishing of some derivatives of the observable at fixed points. The proof of this new phenomenon is inspired by tools developed in the recent work of the first author and Ulcigrai for locally Hamiltonian flows having only (simple) non-degenerate saddles. In full generality, due to the occurrence of (multiple) degenerate saddles, we introduce new methods of handling functions with polynomial singularities over a full measure set of interval exchange transformations.

Measure rigidity of Anosov flows via the factorization method

Using the factorization method, a method pioneered by Eskin and Mirzakhani in their groundbreaking work about measure classification over the moduli space of translation surfaces, we show that generalized $u$-Gibbs states over quantitatively non-integrable Anosov systems are absolutely continuous with respect to the whole unstable manifold.

A criterion for density of the isoperiodic leaves in rank one affine invariant orbifolds

AbstractWe define on any affine invariant orbifold a foliation that generalizes the isoperiodic foliation on strata of the moduli space of translation surfaces and study the dynamics of its leaves in the rank 1 case. We establish a criterion that ensures the density of the leaves and provide two applications of this criterion. The first one is a classification of the dynamical behavior of the leaves of when is a connected component of a Prym eigenform locus in genus 2 or 3 and the second provides the first examples of dense isoperiodic leaves in the stratum .

Rank 2 affine manifolds in genus $3$

We complete the classification of rank two affine manifolds in the moduli space of translation surfaces in genus three. Combined with a recent result of Mirzakhani and Wright, this completes the classification of higher rank affine manifolds in genus three.

An origami of genus 3 with arithmetic Kontsevich–Zorich monodromy

AbstractIn this we exploit the arithmeticity criterion of Oh and Benoist–Miquel to exhibit an origami in the principal stratum of the moduli space of translation surfaces of genus three whose Kontsevich–Zorich monodromy is not thin in the sense of Sarnak.

Counting saddle connections in a homology class modulo <inline-formula><tex-math id="M1">\begin{document}$ \boldsymbol q $\end{document}</tex-math></inline-formula> (with an appendix by Rodolfo Gutiérrez-Romo)

We give effective estimates for the number of saddle connections on a translation surface that have length \begin{document}$ \leq L $\end{document} and are in a prescribed homology class modulo \begin{document}$ q $\end{document} . Our estimates apply to almost all translation surfaces in a stratum of the moduli space of translation surfaces, with respect to the Masur–Veech measure on the stratum.

Classification of Rauzy–Veech groups: proof of the Zorich conjecture

We classify the Rauzy-Veech groups of all connected components of all strata of the moduli space of translation surfaces in absolute homology, showing, in particular, that they are commensurable to arithmetic lattices of symplectic groups. As a corollary, we prove a conjecture of Zorich about the Zariski-density of such groups.

The Lagrange spectrum of some square-tiled surfaces

Lagrange spectra have been defined for closed submanifolds of the moduli space of translation surfaces which are invariant under the action of SL(2, R). We consider the closed orbit generated by a specific covering of degree 7 of the standard torus, which is an element of the stratum H(2). We give an explicit formula for the values in the spectrum, in terms of a cocycle over the classical continued fraction. Differently from the classical case of the modular surface, where the lowest part of the Lagrange spectrum is discrete, we find an isolated minimum, and a set with a rich structure right above it.

Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions

In this paper we prove results on Birkhoff and Oseledets genericity along certain curves in the space of affine lattices and in moduli spaces of translation surfaces. In the space of affine lattices \begin{document}$ASL_2( \mathbb{R})/ASL_2( \mathbb{Z})$\end{document} , we prove that almost every point on a curve with some non-degeneracy assumptions is Birkhoff generic for the geodesic flow. This implies almost everywhere genericity for some curves in the locus of branched covers of the torus inside the stratum \begin{document}$\mathscr{H}(1,1)$\end{document} of translation surfaces. For these curves we also prove that almost every point is Oseledets generic for the Kontsevitch-Zorich cocycle, generalizing a recent result by Chaika and Eskin. As applications, we first consider a class of pseudo-integrable billiards, billiards in ellipses with barriers, and prove that for almost every parameter, the billiard flow is uniquely ergodic within the region of phase space in which it is trapped. We then consider any periodic array of Eaton retroreflector lenses, placed on vertices of a lattice, and prove that in almost every direction light rays are each confined to a band of finite width. Finally, a result on the gap distribution of fractional parts of the sequence of square roots of positive integers is also obtained.

Logarithmic laws and unique ergodicity

We show that Masur's logarithmic law of geodesics in the moduli space of translation surfaces does not imply unique ergodicity of the translation flow, but that a similar law involving the flat systole of a Teichmuller geodesic does imply unique ergodicity. It shows that the flat geometry has a better control on ergodic properties of translation flow than hyperbolic geometry.

Everything is illuminated

We study geometrical properties of translation surfaces: the finite blocking property, bounded blocking property, and illumination properties. These are elementary properties which can be fruitfully studied using the dynamical behavior of the SL(2,R)-action on the moduli space of translation surfaces. We characterize surfaces with the finite blocking property and bounded blocking property, completing work of the second-named author. Concerning the illumination problem, we also extend results of Hubert-Schmoll-Troubetzkoy, removing the hypothesis that the surface in question is a lattice surface, thus settling a conjecture. Our results crucially rely on the recent breakthrough results of Eskin-Mirzakhani and Eskin-Mirzakhani-Mohammadi, and on related results of Wright.

Ergodic theory and Diophantine approximation for translation surfaces and linear forms

We derive results on the distribution of directions of saddle connections on translation surfaces using only the Birkhoff ergodic theorem applied to the geodesic flow on the moduli space of translation surfaces. Our techniques, together with an approximation argument, also give an alternative proof of a weak version of a classical theorem in multi-dimensional Diophantine approximation due to Schmidt (1960 Can. J. Math. 12 619–31, 1964 Trans. Am. Math. Soc. 110 493–518). The approximation argument allows us to deduce the Birkhoff genericity of almost all lattices in a certain submanifold of the space of unimodular lattices from the Birkhoff genericity of almost all lattices in the whole space and similarly for the space of affine unimodular lattices.

Counting invariant components of hyperelliptic translation surfaces

The flow in a fixed direction on a translation surface S determines a decomposition of S into closed invariant sets, each of which is either periodic or minimal. We study this decomposition for translation surfaces in the hyperelliptic connected components Hhyp(2g − 2) and Hhyp(g − 1, g − 1) of the corresponding strata of the moduli space of translation surfaces. Specifically, we characterize the pairs of nonnegative integers (p,m) for which there exists a translation surface in Hhyp(2g−2) or Hhyp(g−1, g−1) with precisely p periodic components and m minimal components. This extends results by Naveh ([Nav08]), who obtained tight upper bounds on numbers of invariant components for each stratum.

Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards

This text is an expanded version of the lecture notes of a minicourse (with the same title of this text) delivered by the authors in the Bedlewo school ``Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory'' (from 4 to 16 July, 2011). &nbsp In the first part of this text, i.e., from Sections 1 to 5, we discuss the Teichmuller and moduli space of translation surfaces, the Teichmuller flow and the $SL(2,\mathbb{R})$-action on these moduli spaces and the Kontsevich--Zorich cocycle over the Teichmuller geodesic flow. We sketch two applications of the ergodic properties of the Teichmuller flow and Kontsevich--Zorich cocycle, with respect to Masur--Veech measures, to the unique ergodicity, deviation of ergodic averages and weak mixing properties of typical interval exchange transformations and translation flows. These applications are based on the fundamental fact that the Teichmuller flow and the Kontsevich--Zorich cocycle work as renormalization dynamics for interval exchange transformations and translation flows. &nbsp In the second part, i.e., from Sections 6 to 9, we start by pointing out that it is interesting to study the ergodic properties of the Kontsevich--Zorich cocycle with respect to invariant measures other than the Masur--Veech ones, in view of potential applications to the investigation of billiards in rational polygons (for instance). We then study some examples of measures for which the ergodic properties of the Kontsevich--Zorich cocycle are very different from the case of Masur--Veech measures. Finally, we end these notes by constructing some examples of closed $SL(2,\mathbb{R})$-orbits such that the restriction of the Teichmuller flow to them has arbitrary small rate of exponential mixing, or, equivalently, the naturally associated unitary $SL(2,\mathbb{R})$-representation has arbitrarily small spectral gap (and in particular it has complementary series).

Lagrange Spectra in Teichmüller Dynamics via Renormalization

We introduce Lagrange Spectra of closed-invariant loci for the action of SL(2, $${\mathbb{R}}$$ ) on the moduli space of translation surfaces, generalizing the classical Lagrange Spectrum, and we analyze them with renormalization techniques. A formula for the values in such spectra is established in terms of the Rauzy–Veech induction and it is used to show that any invariant locus has closed Lagrange spectrum and values corresponding to pseudo-Anosov elements are dense. Moreover we show that Lagrange spectra of arithmetic Teichmüller discs contain an Hall’s ray, giving an explicit bound for it via a second formula which uses the classical continued fraction algorithm. In addition, we show the equivalence of several definitions of bounded Teichmüller geodesics and bounded type interval exchange transformations and we prove quantitative estimates on excursions to the boundary of moduli space in terms of norms of positive matrices in the Rauzy–Veech induction.

Labeled Rauzy classes and framed translation surfaces

In this paper, we compare two definitions of Rauzy classes. The first one was introduced by Rauzy and was in particular used by Veech to prove the ergodicity of the Teichmuller flow. The second one is more recent and uses a labeling of the underlying intervals, and was used in the proof of some recent major results about the Teichmuller flow. The Rauzy diagrams obtained from the second definition are coverings of the initial ones. In this paper, we give a formula that gives the degree of this covering. This formula is related to moduli spaces of framed translation surfaces, which corresponds to surfaces where we label horizontal separatrices on the surface. We compute the number of connected component of these natural coverings of the moduli spaces of translation surfaces. Delecroix proved recently a recursive formula for the cardinality of the (reduced) Rauzy classes. Therefore, we also obtain a formula for labeled Rauzy classes.

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

We present a Galois-theoretical criterion for the simplicity of the Lyapunov spectrum of the Kontsevich-Zorich cocycle over the Teichmueller 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 $\geq 3$. We apply our criterion to square-tiled surfaces of genus 3 with one single zero. Conditionally to a conjecture of Delecroix and Leli\`evre, 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.

Cohomology classes represented by measured foliations, and Mahler's question for interval exchanges

A translation surface on (S;�) gives rise to two transverse measured foliations F;G on S with singularities in �, and by integration, to a pair of cohomology classes (F);(G) 2 H 1 (S;�;R). Given a measured foliation F, we characterize the set of cohomology classes b for which there is a measured foliation G as above with b = (G). This extends previous results of Thurston (Th) and Sullivan (Su). We apply this to two problems: unique ergodicity of interval ex- changes and flows on the moduli space of translation surfaces. For a fixed permutation � 2 Sd, the space R d parametrizes the interval ex- changes on d intervals with permutation �. We describe linesin R d such that almost every point inis uniquely ergodic. We also show that for �(i) = d+1 i, for almost every s > 0, the interval exchange trans- formation corresponding toand (s;s 2 ;:::;s d ) is uniquely ergodic. As another application we show that when k = |�| � 2;the operation of 'moving the singularities horizontally' is globally well-defined. We prove that there is a well-defined action of the group B n R k 1 on the set of translation surfaces of type (S;�) without horizontal saddle connections. Here BSL(2;R) is the subgroup of upper triangular matrices.

$${SL(2, \mathbb{R})}$$ S L ( 2 , R ) -invariant probability measures on the moduli spaces of translation surfaces are regular

In the moduli space \({{\mathcal {H}}_g}\) of normalized translation surfaces of genus g, consider, for a small parameter ρ > 0, those translation surfaces which have two non-parallel saddle-connections of length ⩽ ρ. We prove that this subset of \({{\mathcal {H}}_g}\) has measure o(ρ2) w.r.t. any probability measure on \({{\mathcal {H}}_g}\) which is invariant under the natural action of \({SL(2,\mathbb{R})}\) . This implies that any such probability measure is regular, a property which is important in relation with the recent fundamental work of Eskin–Kontsevich–Zorich on the Lyapunov exponents of the KZ-cocycle.

Labeled Rauzy classes and framed translation surfaces

In this paper, we compare two definitions of Rauzy classes. The first one was introduced by Rauzy and was in particular used by Veech to prove the ergodicity of the Teichmuller flow. The second one is more recent and uses a labeling of the underlying intervals, and was used in the proof of some recent major results about the Teichmuller flow. The Rauzy diagrams obtained from the second definition are coverings of the initial ones. In this paper, we give a formula that gives the degree of this covering. This formula is related to moduli spaces of framed translation surfaces, which corresponds to surfaces where we label horizontal separatrices on the surface. We compute the number of connected component of these natural coverings of the moduli spaces of translation surfaces. Delecroix proved recently a recursive formula for the cardinality of the (reduced) Rauzy classes. Therefore, we also obtain a formula for labeled Rauzy classes.