Schedule for: 19w5078 - Flat Surfaces and Dynamics on Moduli Space, II

Abstract: Introduced by Furstenberg, joinings between two transformations (X,mu,T) and (Y,nu,S) are TxS invariant measures on XxY that project on the coordinates to mu and nu respectively. They describe the ways that these transformations 'talk' to each other. In particular, a common factor of these transformations gives rise to a joining between them. This talk will survey, known results on joinings between different interval exchange transformations and flows on translation surfaces, different powers of the same transformations and self-joinings. No familiarity with joinings will be assumed.

We will begin by defining the rank of an affine invariant submanifold in a stratum of abelian differentials on genus g surfaces. We will end by showing that if the rank is at least g/2 + 1 then the affine invariant submanifold is either the full component of the stratum containing it or a holonomy double cover of a component of a stratum of quadratic differentials. This is joint work with Alex Wright.

I will present a formula for the Masur-Veech volumes of the principal strata of quadratic differentials (as well as Siegel-Veech constants) in terms of intersection numbers involving psi-classes. This formula closely relates the counting of square-tiled surfaces with Mirzakhani's counting of simple closed geodesic multicurves on hyperbolic surfaces, and leads to several conjectures for the large genus asymptotics. This is a joint work with V.Delecroix, P.Zograf and A.Zorich.

Classification of proper non arithmetic affine manifolds in H(2,2)^{odd} and H(3,1)

For a half-translation surface, the saddle connection complex is the induced subcomplex of the arc complex where vertices are saddle connections. Every simplicial isomorphism between the saddle connection complexes of two half-translation surfaces is induced by an affine diffeomorphism between the underlying surfaces. In this talk, I will explain some properties of the saddle connection complex and outline the proof of the mentioned rigidity result. Both is based on joint work with Valentina Disarlo and Robert Tang.

A collection of polygons with the property that to each side one can find another side parallel to it can be endowed with a translation surface structure by glueing along those edges. This means that the closed surfaces obtained carries a flat metric outside finitely many conical singularities. Geodesics (which are straight lines) connecting such singularities are called saddle connections. While the asymptotic number of saddle connections of length less then T growth roughly like T^2 (in the sense that there are lower and upper bounds of that order), one can say more for a generic surface with respect to the moduli space of such structures thanks to the natural SL2-action it is equipped with. I shall present some results with polynomial error saving for counting saddle connections in the setting of a) general loci (j/w Nevo,Weiss) b) prescribed congruence restrictions in homology (j/w Magee, Guetierrez-Romo) c) lattice-surfaces using Eisenstein series (j/w Burrin,Nevo,Weiss)

A group acting on an elliptic curve must have order N = 1, 2, 3, 4, or 6. We call the quotient an elliptic orbifold. Certain branched covers of the order N elliptic orbifold are in bijection with tiled surfaces, and form a lattice in the moduli space of N-ic differentials on Riemann surfaces. The enumerative theory of these branched covers suggests a phantom "elliptic orbifold" for all integers N. I will discuss work-in-progress with Peter Smillie proposing a definition for the Hurwitz theory of this non-existent object, and attempts to relate it to quasi-crystals in the moduli space of quintic differentials and the enumeration of Penrose-tiled Riemann surfaces.

Problem session

Let X be a closed, connected, hyperbolic surface of genus 2. Is it more likely for a simple closed geodesic on X to be separating or non-separating? How much more likely? In her thesis, Mirzakhani gave very precise answers to these questions. One can ask analogous questions for square-tiled surfaces of genus 2 with one horizontal cylinder. Is it more likely for such a square-tiled surface to have separating or non-separating horizontal core curve? How much more likely? Recently, Delecroix, Goujard, Zograf, and Zorich gave very precise answers to these questions. Surprisingly enough, their answers were exactly the same as the ones in Mirzakhani’s work. In this talk we explore the connections between these counting problems, showing they are related by more than just an accidental coincidence.

Kontsevich and Zorich famously classified the connected components of strata of translation surfaces over moduli space. The corresponding problem over the Teichmüller space requires the analysis of which mapping classes can be realized as deformations lying inside the stratum. In this talk, I will present joint work with Nick Salter in which we classify the (non-hyperelliptic) connected components of strata over Teichmüller space. Unlike the case for strata over moduli space, we find that there can be many (but finitely many) connected components, depending on both genus and cone angle.

An IET is rational if it translates points by rational amount. In a finite rational IET, all orbits are periodic. In an infinite IETs, this need not be the case. I will discuss some examples of infinite rational IETs in which it can be proved that almost every point is periodic. Further the dynamics on the aperiodic sets can be completely understood. Theory developed to understand these examples points out an interesting class of infinite IETs admitting a renormalization scheme (though it remains to be seen how effective this scheme is for understanding generic IETs). I will be discussing a preprint joint with Kasra Rafi and Anja Randecker, and a work in progress joint with Anna Tao.

The Gauss circle problem consists in counting the number of integer points of bounded length in the plane. This problem is equivalent to counting the number of closed geodesics of bounded length on a flat two dimensional torus. Many counting problems in dynamical systems have been inspired by this problem. For 30 years, the experts try to understand the asymptotic behavior of closed geodesics in translation surfaces and periodic trajectories on rational billiards. (Polygonal billiards yield translation surfaces naturally through an unfolding procedure.) H. Masur proved that this number has quadratic growth rate. In this talk, we will study the counting problem on infinite periodic rational billiards and translation surfaces. The first example and motivation is the wind-tree model, a Z^2-periodic billiard model. In the classical setting, we place identical rectangular obstacles in the plane at each integer point; we play billiard on the complement. It is possible to give quite precise results on the counting problem for this model, thanks to the many symmetries it presents. These results, however, do not extend to more general contexts. I will present a general result on the counting problem for infinite periodic translation surfaces that uses new ideas: a dynamical analogous, for the algebraic hull of a cocycle, to strong and super-strong approximation on algebraic groups. Under these approximation hypothesis I will exhibit asymptotic formulas for the number of closed geodesics of bounded length on infinite periodic translation surfaces. And will present some applications and discuss why I think these hypothesis hold in general (work in progress).

When geometric structures on surfaces are determined by the lengths of curves, it is natural to ask which curves’ lengths do we really need to know? It is a classical result of Fricke that a hyperbolic metric on a surface is determined by its marked simple length spectrum. More recently, Duchin–Leininger–Rafi proved that a flat metric induced by a unit-norm quadratic differential is also determined by its marked simple length spectrum. In this talk, I will describe a generalization of the notion of simple curves to that of q-simple curves, for any positive integer q, and show that the lengths of q-simple curves suffice to determine a non-positively curved Euclidean cone metric induced by a q-differential metric.

There is a natural forgetful map p from a stratum H of abelian differentials to the moduli space of Riemann surfaces that takes a pair (X,w) to X. What are the coarse properties of this map p? In particular, when is the image coarsely dense with respect to the Teichmuller metric on moduli space? We answer this question for all strata by showing that the projection is coarsely dense iff it is topologically dense. The proof uses a new compactification of strata due to Bainbridge-Chen-Gendron-Grushevsky-Moller. This is joint work with Jenya Sapir.

In this talk I'll explain how the shape of a Euclidean polygon is essentially determined by the symbolic coding of the "generalized diagonal" billiard trajectories. The bulk of the talk will involve the reduction of this theorem to a rigidity result for flat surfaces. This is joint work with Moon Duchin, Viveka Erlandsson, and Chandrika Sadanand.