On the Space of Ergodic Measures for the Horocycle Flow on Strata of Abelian Differentials
Jon Chaika, Osama Khalil, John Smillie
备注：
Corrections based on referee reports, clarifications of statements of main results and of the arguments in Section 7, expanded and clarified some of the discussions in the introduction. Main results are not affected
We study the horocycle flow on the stratum of translation surfaces $\mathcal{H}(2)$. We show that there is a sequence of horocycle ergodic measures, each supported on a periodic horocycle orbit, which weakly converges to an invariant, but non-ergodic, measure by $\mathrm{SL}_2(\mathbb{R})$. As a consequence, we show that there are points in $\mathcal{H}(2)$ whose horocycle flow orbits do not equidistribute towards any invariant measure.
We investigate specific examples of locally-defined real vector-fields on strata of translation surfaces. Integrating SL(2,R)-loci of Veech surfaces along these vector-fields yield interesting new examples of horocyle-invariant ergodic measures. These measures are supported on closed immersed manifolds with boundary that approach the boundary of strata in a novel way.
A classic result due to Furstenberg is the strict ergodicity of the horocycle flow for a compact hyperbolic surface. Strict ergodicity is unique ergodicity with respect to a measure of full support, and therefore implies minimality. The horocycle flow has been previously studied on minimal foliations by hyperbolic surfaces on closed manifolds, where it is known not to be minimal in general. In this paper, we prove that for the special case of Riemannian foliations, strict ergodicity of the horocycle flow still holds. This in particular proves that this flow is minimal, which establishes a conj…
We show that the horocyclic flow of an orientable compact higher genus surface without conjugate points and with continuous Green bundles is uniquely ergodic. The result applies to nonflat nonpositively curved surfaces and generalizes a classical result of Furstenberg and Marcus in negative curvature. The proof relies on the definition of a uniformly expanding parametrization on the quotient by the strips of the surface.
We apply a method inspired by Ratner's work on quantitative mixing for the geodesic flow (Ergod. Theory Dyn. Syst., 1987) and developed by Burger (Duke Math. J., 1990) to study ergodic integrals for horocycle flows. We derive an explicit asymptotic expansion for horocycle averages, recovering a celebrated result by Flaminio and Forni (Duke Math. J., 2003), and we show that the coefficients in the asymptotic expansion are Hölder continuous with respect to the base point. Furthermore, we provide short and streamlined proofs of the spatial limit theorems of Bufetov and Forni (Ann. Sci. Éc. Norm. …
We study dynamics of the horocycle flow on strata of translation surfaces, introduce new invariants for ergodic measures, and analyze the interaction of the horocycle flow and real Rel surgeries. We use this analysis to complete and extend results of Calta and Wortman classifying horocycle-invariant measures in the eigenform loci. We classify the orbit-closures and prove that every orbit is equidistributed in its orbit-closure. We also prove equidistribution statements regarding limits of sequences of measures, some of which have applications to counting problems.
A point is called generic for a flow preserving an infinite ergodic invariant Radon measure, if its orbit satisfies the conclusion of the ratio ergodic theorem for every pair of continuous functions with compact support and non-zero integrals. The generic points for horocycle flows on hyperbolic surfaces of finite genus are understood, but there are no results in infinite genus. We give such a result, by characterizing the generic points for $\Z^d$--covers.
We show that the ergodic averages for the horocycle flow on the two-torus associated by Giulietti and Liverani to an Anosov diffeomorphism either grow linearly or are bounded, in other words there are no deviations. For this, we use topological invariance of the Artin-Mazur zeta function to exclude resonances outside of the open unit disc. Transfer operators acting on suitable spaces of anisotropic distributions and their Ruelle determinants are the key tools in the proof. As a bonus, we show that for any smooth Anosov diffeomorphism F on the two-torus, the correlations for the measure of maxi…
We show that the Hausdorff dimension of any proper Teichmüller horocycle flow orbit closure on any $\mathrm{SL}{(2,\mathbf{R})}$-invariant subvariety of Abelian or quadratic differentials is bounded away from the dimension of the subvariety in terms of the polynomial mixing rate of the Teichmüller horocycle flow on the subvariety. The proof is based on abstract methods for measurable flows adapted from work of Bourgain and Katz on sparse ergodic theorems.
We prove bounds for twisted ergodic averages for horocycle flows of hyperbolic surfaces, both in the compact and in the non-compact finite area case. From these bounds we derive effective equidistribution results for horocycle maps. As an application of our main theorems in the compact case we further improve on a result of A. Venkatesh, recently already improved by J. Tanis and P. Vishe, on a sparse equidistribution problem for classical horocycle flows proposed by N. Shah and G. Margulis, and in the general non-compact, finite area case we prove bounds on Fourier coefficients of cups forms w…
We construct a Poincaré section for the horocycle flow on the modular surface $SL(2, \R)/SL(2, \Z)$, and study the associated first return map, which coincides with a transformation (the {\it BCZ map}) defined by Boca-Cobeli-Zaharescu. We classify ergodic invariant measures for this map and prove equidistribution of periodic orbits. As corollaries, we obtain results on the average depth of cusp excursions and on the distribution of gaps for Farey sequences and slopes of lattice vectors.
We adapt to $C^r$ Anosov flows on compact manifolds a construction for $C^r$ discrete-time hyperbolic dynamics ($r>1$), obtaining anisotropic Banach or Hilbert spaces on which the resolvent of the generator of weighted transfer operators for the flow is quasi-compact. We apply this to study the ergodic integrals of the horocycle flows $h_ρ$ of $C^r$ codimension one mixing Anosov flows. In dimension three, for any suitably bunched $C^3$ contact Anosov flow with orientable strong-stable distribution, we establish power-law convergence of the ergodic average. We thereby implement the program of G…
The main results of this paper are limit theorems for horocycle flows on compact surfaces of constant negative curvature. One of the main objects of the paper is a special family of horocycle-invariant finitely-additive Hoelder measures on rectifiable arcs. An asymptotic formula for ergodic integrals for horocycle flows is obtained in terms of the finitely-additive measures, and limit theorems follow as a corollary of the asymptotic formula. The objects and results of this paper are similar to those in [15], [16], [4] and [5] for translation flows on flat surfaces. The arguments are based on t…
We present a method, based on commutator methods, for the spectral analysis of uniquely ergodic dynamical systems. When applicable, it leads to the absolute continuity of the spectrum of the corresponding unitary operators. As an illustration, we consider time changes of horocycle flows, skew products over translations and Furstenberg transformations. For time changes of horocycle flows, we obtain absolute continuity under assumptions weaker than the ones to be found in the literature.
We study the action of the horocycle flow on the moduli space of abelian differentials in genus two. In particular, we exhibit a classification of a specific class of probability measures that are invariant and ergodic under the horocycle flow on the stratum H(1,1).
In two fundamental classical papers, Masur and Veech have independently proved that the Teichmueller geodesic flow acts ergodically on each connected component of each stratum of the moduli space of quadratic differentials. It is therefore interesting to have a classification of the ergodic components. Veech has proved that these strata are not necessarily connected. In a recent work, Kontsevich and Zorich have completely classified the components in the particular case where the quadratic differentials are given by the global square of Abelian differentials. Here we are interested in the comp…
Let $M$ be a closed oriented surface endowed with a Riemannian metric $g$ and let $Ω$ be a 2-form. We show that the magnetic flow of the pair $(g,Ω)$ has zero asymptotic Maslov index and zero Liouville action if and only $g$ has constant Gaussian curvature, $Ω$ is a constant multiple of the area form of $g$ and the magnetic flow is a horocycle flow. This characterization of horocycle flows implies that if the magnetic flow of a pair $(g,Ω)$ is $C^1$-conjugate to the horocycle flow of a hyperbolic metric $\bar{g}$ then there exists a constant $a>0$, such that $ag$ and $\bar{g}$ are isometric an…
The Ratner property, a quantitative form of divergence of nearby trajectories, is a central feature in the study of parabolic homogeneous flows. Discovered by Marina Ratner and used in her 1980th seminal works on horocycle flows, it pushed forward the disjointness theory of such systems. In this paper, exploiting a recent variation of the Ratner property, we prove new disjointness phenomena for smooth parabolic flows beyond the homogeneous world. In particular, we establish a general disjointness criterion based on the switchable Ratner property. We then apply this new criterion to study disjo…
We study the ergodic properties of horospheres on rank 1 manifolds with non-positive curvature. We prove that the horospheres are equidistributed under the action of the geodesic flow towards the Bowen-Margulis measure, on a large class of manifolds. In the case of surfaces, we define a parametrization of the horocyclic flow on the set of horocycles containing a rank 1 vector that is recurrent under the action of the geodesic flow. We prove that the horocyclic flow in restriction to this set is uniquely ergodic. The results are valid for large classes of manifolds, including the compact ones.
Moduli spaces of Abelian and quadratic differentials are stratified by multiplicities of zeroes; connected components of the strata correspond to ergodic components of the Teichmuller geodesic flow. It is known that the strata are not necessarily connected; the connected components were recently classified by M. Kontsevich and the author and by E. Lanneau. The strata can be also viewed as families of flat metrics with conical singularities and with Z/2Z-holonomy. For every connected component of each stratum of Abelian and quadratic differentials we construct an explicit representative which i…
We provide a self-contained, accessible introduction to Ratner's Equidistribution Theorem in the special case of horocyclic flow on a complete hyperbolic surface of finite area. This equidistribution result was first obtained in the early 1980s by Dani and Smillie and later reappeared as an illustrative special case of Ratner's work on the equidistribution of unipotent flows in homogeneous spaces. We also prove an interesting probabilistic result due to Breuillard: on the modular surface an arbitrary uncentered random walk on the horocycle through almost any point will fail to equidistribute, …
Ten years ago A. Zorich discovered, by computer experiments on interval exchange transformations, some striking new power laws for the ergodic integrals of generic non-exact Hamiltonian flows on higher genus surfaces. In Zorich's later work and in a joint paper authored by M. Kontsevich, Zorich and Kontsevich were able to explain conjecturally most of Zorich's discoveries by relating them to the ergodic theory of Teichmüller flows on moduli spaces of Abelian differentials. In this article, we outline a generalization of the Kontsevich-Zorich framework to a class of `renormalizable' flows on `p…
We consider the action of $SL(2,\mathbb{R})$ on a vector bundle $\mathbf{H}$ preserving an ergodic probability measure $ν$ on the base $X$. Under an irreducibility assumption on this action, we prove that if $\hatν$ is any lift of $ν$ to a probability measure on the projectivized bunde $\mathbb{P}(\mathbf{H})$ that is invariant under the upper triangular subgroup, then $\hat ν$ is supported in the projectivization $\mathbb{P}(\mathbf{E}_1)$ of the top Lyapunov subspace of the positive diagonal semigroup. We derive two applications. First, the Lyapunov exponents for the Kontsevich-Zorich cocycl…
We use the relation between the volumes of the strata of meromorphic quadratic differentials with at most simple poles on the Riemann sphere and counting functions of the number of (bands of) closed geodesics in associated flat metrics with singularities to prove a very explicit formula for the volume of each such stratum conjectured by M. Kontsevich a decade ago. The proof is based on a formula for the Lyapunov exponents of the geodesic flow on the moduli space, which gives a recursive formula from which the volumes could be recovered. An appendix with an ergodic theorem proved by Jon Chaika …
We prove a geometric criterion on a $\SL$-invariant ergodic probability measure on the moduli space of holomorphic abelian differentials on Riemann surfaces for the non-uniform hyperbolicity of the Kontsevich--Zorich cocycle on the real Hodge bundle. Applications include measures supported on the $\SL$-orbits of all algebraically primitive Veech surfaces (see also \cite{Bouw:Moeller}) and of all Prym eigenforms discovered in \cite{McMullen2}, as well as all canonical absolutely continuous measures on connected components of strata of the moduli space of abelian differentials (see also \cite{Ft…
We introduce the geodesic flow on the leaves of a holomorphic foliation with leaves of dimension 1 and hyperbolic, corresponding to the unique complete metric of curvature -1 compatible with its conformal structure. We do these for the foliations associated to Riccati equations, which are the projectivisation of the solutions of linear ordinary differential equations over a finite Riemann surface of hyperbolic type S, and may be described by a representation r:pi_1(S) -> GL(n,C). We give conditions under which the foliated geodesic flow has a generic repellor-attractor statistical dynamics. Th…
