On the space of ergodic measures for the horocycle flow on strata of Abelian differentials

We compute many new classes of effective divisors in M¯g,n coming from the strata of Abelian differentials. Our method utilizes maps between moduli spaces and the degeneration of Abelian differentials.

We study the transcendence of periods of abelian differentials, both at the arithmetic and functional level, from the point of view of the natural bi-algebraic structure on strata of abelian differentials. We characterize geometrically the arithmetic points, study their distribution, and prove that in many cases the bi-algebraic curves are the linear ones.

This paper focuses on the interplay between the intersection theoryand the Teichmüller dynamics on the moduli space of curves. Asapplications, we study the cycle class of strata of the Hodge bundle,present an algebraic method to calculate the class of the divisorparameterizing abelian differentials with a nonsimple zero, andverify a number of extremal effective divisors on the moduli space ofpointed curves in low genus.

An abelian differential on a surface defines a flat metric and a vector field on the complement of a finite set of points. The vertical flow that can be defined on the surface has two kinds of invariant closed sets (i.e. invariant components) — periodic components and minimal components. We give upper bounds on the number of minimal components, on the number of periodic components and on the total number of invariant components in every stratum of abelian differentials. We also show that these bounds are tight in every stratum.

Cette note donne une preuve élémentaire que les strates des différentiels abéliens ne contiennent pas de variétés algébriques complètes.

We show that an algebraic subvariety of the moduli space of genus g Riemann surfaces is coarsely dense with respect to the Teichmüller metric (or Thurston metric) if and only if it has full dimension. We apply this to determine which strata of abelian differentials have coarsely dense projection to moduli space. Furthermore, we prove a result on coarse density of projections of GL 2 (ℝ)-orbit closures in the space of abelian differentials.

A cyclic cover of the complex projective line branched at four appropriate points has a natural structure of a square-tiled surface. We describe the combinatorics of such a square-tiled surface, the geometry of the corresponding Teichmüller curve, and compute the Lyapunov exponents of the determinant bundle over the Teichmüller curve with respect to the geodesic flow. This paper includes a new example (announced by G. Forni and C. Matheus in [17] of a Teichmüller curve of a square-tiled cyclic cover in a stratum of Abelian differentials in genus four with a maximally degenerate Kontsevich--Zorich spectrum (the only known example in genus three found previously by Forni also corresponds to a square-tiled cyclic cover [15]. We present several new examples of Teichmüller curves in strata of holomorphic and meromorphic quadratic differentials with a maximally degenerate Kontsevich--Zorich spectrum. Presumably, these examples cover all possible Teichmüller curves with maximally degenerate spectra. We prove that this is indeed the case within the class of square-tiled cyclic covers.

We show that for many strata of Abelian differentials in low genus the sum of Lyapunov exponents for the Teichmuller geodesic flow is the same for all Teichmuller curves in that stratum, hence equal to the sum of Lyapunov exponents for the whole stratum. This behavior is due to the disjointness property of Teichmuller curves with various geometrically defined divisors on moduli spaces of curves. 14H10; 37D40, 14H51

We show that the Masur–Veech volumes and area Siegel–Veech constants can be obtained using intersection theory on strata of Abelian differentials with prescribed orders of zeros. As applications, we evaluate their large genus limits and compute the saddle connection Siegel–Veech constants for all strata. We also show that the same results hold for the spin and hyperelliptic components of the strata.

We introduce a twisted cohomology cocycle over the Teichmüller flow and prove a “spectral gap” for its Lyapunov spectrum with respect to the Masur–Veech measures. We then derive Hölder estimates on spectral measures and bounds on the speed of weak mixing for almost all translation flows in every stratum of Abelian differentials on Riemann surfaces, as well as bounds on the deviation of ergodic averages for product translation flows on the product of a translation surface with a circle.

On the effective cone of [formula omitted

It is well known that the space of invariant probability measures for transitive sub-shifts of finite type is a Poulsen simplex. In this article we prove that in the non-compact setting, for a large family of transitive countable Markov shifts, the space of invariant sub-probability measures is a Poulsen simplex and that its extreme points are the ergodic invariant probability measures together with the zero measure. In particular, we obtain that the space of invariant probability measures is a Poulsen simplex minus a vertex and the corresponding convex combinations. Our results apply to finite entropy non-locally compact transitive countable Markov shifts and to every locally compact transitive countable Markov shift. In order to prove these results we introduce a topology on the space of measures that generalizes the vague topology to a class of non-locally compact spaces, the topology of convergence on cylinders. We also prove analogous results for suspension flows defined over countable Markov shifts.

For dynamical systems with the shadowing property, we provide a method of approximation of invariant measures by ergodic measures supported on odometers and their almost one-to-one extensions. For a topologically transitive system with the shadowing property, we show that ergodic measures supported on odometers are dense in the space of invariant measures, and then ergodic measures are generic in the space of invariant measures. We also show that for every $c\geq 0$ and $\unicode[STIX]{x1D700}>0$ the collection of ergodic measures (supported on almost one-to-one extensions of odometers) with entropy between $c$ and $c+\unicode[STIX]{x1D700}$ is dense in the space of invariant measures with entropy at least $c$. Moreover, if in addition the entropy function is upper semi-continuous, then, for every $c\geq 0$, ergodic measures with entropy $c$ are generic in the space of invariant measures with entropy at least $c$.

Contents Introduction §0. Agreements, definitions and preliminaries §1. Surfaces of constant negative curvature §2. Measure rigidity of the horocycle flow §3. Geometric generalizations of Ratner's theorem on measure rigidity of the horocycle flow §4. Quotients and joinings of the horocycle flow §5. Rigidity, quotients and joinings of unipotent flows §6. Dynamics of the horocycle flow §7. Classification of ergodic measures for unipotent flows §8. Uniform distribution of unipotent trajectories §9. Various problems of convergence in the space of measures §10. Structure of orbits, minimal sets, and ergodic measures of homogeneous flows §11. Multiple mixing and measure rigidity of homogeneous flows §12. Ergodic measures and orbit closures for actions of arbitrary subgroups §13. Unipotent flows on homogeneous spaces over local fields §14. Applications to number theory §15. Some open problemsBibliography

We provide an elementary proof that ergodic measures on one-sided shift spaces are ‘uniformly scaling’ in the following sense: at almost every point the scenery distributions weakly converge to a common distribution on the space of measures. Moreover, we show how the limiting distribution can be expressed in terms of, and derived from, a 'reverse Jacobian’ function associated with the corresponding measure on the space of left infinite sequences. Finally we specialise to the setting of Gibbs measures, discuss some statistical properties, and prove a Central Limit Theorem for ergodic Markov measures.