Ergodic theorems for laminations and foliations: recent results and perspectives
Viet-Anh Nguyen
备注：
Presentation improved, Subsections 2.6 and 7.2 revised. 95 pages, 6 figures. This paper is an expanded version of my talk presented at the Vietnam Institute for Advanced Studies in Mathematics (VIASM) at the Annual Meeting on August 17, 2019. arXiv admin note: text overlap with arXiv:1712.09739
This report discusses recent results as well as new perspectives in the ergodic theory for Riemann surface laminations, with an emphasis on singular holomorphic foliations by curves. The central notions of these developments are leafwise Poincaré metric, directed positive harmonic currents, multiplicative cocycles and Lyapunov exponents. We deal with various ergodic theorems for such laminations: Random and Operator Ergodic Theorems, (Geometric) Birkhoff Ergodic Theorems, Oseledec Multiplicative Ergodic Theorem and Unique Ergodicity Theorems. Applications of these theorems are also given. In particular, we define and study the canonical Lyapunov exponents for a large family of singular holomorphic foliations on compact projective surfaces. Topological and algebro-geometric interpretations of these characteristic numbers are also treated.
These results highlight the strong similarity as well as the fundamental differences between the ergodic theory of maps and that of Riemann surface laminations. Most of the results reported here are known. However, sufficient conditions for abstract heat diffusions to coincide with the leafwise heat diffusions (Subsection 5.2) are new ones.
This essay summarizes the state of the art on some aspects of the dynamics of polynomial diffeomorphsms in complex dimension two, and it presents a number of open questions.
In these introductory notes we give the basics of the theory of holomorphic foliations and laminations. The emphasis is on the theory of harmonic currents and unique ergodicity for laminations transversally Lipschitz in CP^2 and for generic holomorphic foliations in CP^2.
We survey some recent developments in the ergodic theory for hyperbolic Riemann surface laminations. The emphasis is on singular holomorphic foliations. These results not only illustrate the strong similarity between the ergodic theory of maps and that of Riemann surface laminations, but also indicate the fundamental differences between these two theories.
Consider a Brody hyperbolic foliation by Riemann surfaces with linearizable isolated singularities on a compact complex surface. We show that its hyperbolic entropy is finite. We also estimate the modulus of continuity of the Poincare metric on leaves. The estimate holds for foliations on manifolds of higher dimension.
We introduce the heat equation relative to a positive dd-bar-closed current and apply it to the invariant currents associated with Riemann surface laminations possibly with singularities. The main examples are holomorphic foliations by Riemann surfaces in projective spaces. We prove two kinds of ergodic theorems for such currents: one associated to the heat diffusion and one close to Birkhoff's averaging on orbits of a dynamical system. The heat diffusion theorem with respect to a harmonic measure is also developed for real laminations.
If $\mathcal{L}$ is a laminations with hyperbolic singularities, embedded in a compact homogeneous Kähler surface, without directed closed positive currents. Then, $\mathcal{L}$ has a unique directed positive harmonic current of mass one. This result was previously obtained by Fornaess and Sibony for $\mathbb{CP}^2$ and we generalize it for the rest of compact homogeneous Kähler surfaces.
Given a n-dimensional lamination endowed with a Riemannian metric, we introduce the notion of a multiplicative cocycle of rank d, where n and d are arbitrary positive integers. The holonomy cocycle of a foliation and its exterior powers as well as its tensor powers provide examples of multiplicative cocycles. Next, we define the Lyapunov exponents of such a cocycle with respect to a harmonic probability measure directed by the lamination. We also prove an Oseledec multiplicative ergodic theorem in this context. This theorem implies the existence of an Oseledec decomposition almost everywhere w…
A new notion of cohomology is introduced for MT-spaces, which are measurable and topological spaces whose measurable structure may not agree with the Borel $σ$-algebra of their topology. The main examples of MTspaces are measurable foliations. This is a singular version of the measurable simplicial cohomology defined by Heitsch and Lazarov for foliations and extended by Bermudez for MT-spaces. Basic topics of algebraic topology are adapted, and applications to the theory of foliations are given. Moreover we introduce a new notion of singular L2-cohomology for MT-spaces.
We construct Weil-Petersson (WP) geodesic rays with minimal filling non-uniquely ergodic ending lamination which are recurrent to a compact subset of the moduli space of Riemann surfaces. This construction shows that an analogue of the Masur's criterion for Teichmüller geodesics does not hold for WP geodesics.
Let F be a holomorphic foliation of P^2 by Riemann surfaces. Assume all the singular points of F are hyperbolic. If F has no algebraic leaf, then there is a unique positive harmonic $(1,1)$ current $T$ of mass one, directed by F. This implies strong ergodic properties for the foliation. We also study the harmonic flow associated to the current $T.$
We introduce the so-called BT-category of borelian-topological spaces: it will be a natural frame for a measurable classification of usual foliations and laminations. We focus on the two-dimensional case: borelian laminations by surfaces. We prove two main results: (1) Any borelian lamination by planes is the suspension of a $\Z^2$-action on a Borel space iff this lamination is hyperfinite. (2) Any borelian lamination by surfaces is amenable if parabolic, i.e. if it admits a complex structure parabolic on each leaf. The third result is an improvement of (2) in case of laminations endowed with …
Masur's divergence states that the horizontal foliation of translation surfaces is uniquely ergodic if the geodesic flow is recurrent on the moduli space. This established a relationship between geometrical properties of foliations and the dynamics on the moduli space. In this paper, we extend this theorem to complex torus and Kummer surfaces. We define and calculate horizontal foliations and the corresponding geodesic flow in the moduli space of Kähler metrics and prove that the horizontal foliation is uniquely ergodic if the geodesic flow is recurrent. We also find the a necessary algebraic …
We construct an example of a quadratic differential whose vertical foliation is uniquely ergodic and such that the Teichmuller geodesic determined by the quadratic differential diverges in the moduli space of Riemann surfaces.
Let \Fc be a holomorphic foliation by Riemann surfaces on a compact Kähler surface X. Assume it is generic in the sense that all the singularities are hyperbolic and that the foliation admits no directed positive closed (1,1)-current. Then there exists a unique (up to a multiplicative constant) positive \ddc-closed (1,1)-current directed by \Fc. This is a very strong ergodic property of \Fc. Our proof uses an extension of the theory of densities to a class of non-\ddc-closed currents. A complete description of the cone of directed positive \ddc-closed (1,1)-currents is also given when \Fc admi…
We show that for a closed surface of genus at least 5, or a surface of genus at least 2 with at least one marked point, the set of uniquely ergodic foliations and the set of cobounded foliations is path-connected and locally path-connected.
In this paper we prove that the limit set of any Weil-Petersson geodesic ray with uniquely ergodic ending lamination is a single point in the Thurston compactification of Teichmüller space. On the other hand, we construct examples of Weil-Petersson geodesics with minimal nonuniquely ergodic ending laminations and limit set a circle in the Thurston compactification.
We describe a method for constructing Teichmüller geodesics where the vertical measured foliation $ν$ is minimal but is not uniquely ergodic and where we have a good understanding of the behavior of the Teichmüller geodesic. The construction depends on various parameters, and we show that one can adjust the parameters to ensure that the set of accumulation points of such a geodesic in the Thurston boundary is exactly the set of all possible measured foliations in the homotopy class of $ν$. With further adjustment of the parameters, one can even take $ν$ to be an ergodic measure on a non-unique…
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 consider the geodesic flow defined by periodic Eaton lens patterns in the plane and discover ergodic ones among those. The ergodicity result on Eaton lenses is derived from a result for quadratic differentials on the plane that are pull backs of quadratic differentials on tori. Ergodicity itself is concluded for $\mathbb{Z}^d$-covers of quadratic differentials on compact surfaces with vanishing Lyapunov exponents.
Given a sequence of curves on a surface, we provide conditions which ensure that (1) the sequence is an infinite quasi-geodesic in the curve complex, (2) the limit in the Gromov boundary is represented by a nonuniquely ergodic ending lamination, and (3) the sequence divides into a finite set of subsequences, each of which projectively converges to one of the ergodic measures on the ending lamination. The conditions are sufficiently robust, allowing us to construct sequences on a closed surface of genus $g$ for which the space of measures has the maximal dimension $3g-3$, for example. We also s…
We construct an example of a uniquely ergodic measured foliation on a surface such that the associated translation flow on the orientation double cover is minimal but not uniquely ergodic. We then prove a geometric criterion for the horizontal foliation of a quadratic differential to be uniquely ergodic. The second theorem generalizes a result of Treviño for the horizontal flow on a translation surface, as well as Masur's criterion for unique ergodicity of the horizontal foliation.
In this work we obtain some metric and ergodic properties of $C^{1+}$ partially hyperbolic diffeomorphisms with one-dimensional topological neutral center, mainly regarding the behavior of its center foliation. Based on a trichotomy for the center conditional measures of any invariant ergodic measure, we show that if these conditionals have full support, then the center foliation is leafwise absolutely continuous, the diffeomorphism is Bernoulli in the $C^{1+}$ case, and an invariance principle occurs in the sense that M may be covered by a finite number of open sets where the system of center…
Let $\mathcal{H}$ be a stratum of translation surfaces with at least two singularities, let $m_{\mathcal{H}}$ denote the Masur-Veech measure on $\mathcal{H}$, and let $Z_0$ be a flow on $(\mathcal{H}, m_{\mathcal{H}})$ obtained by integrating a Rel vector field. We prove that $Z_0$ is mixing of all orders, and in particular is ergodic. We also characterize the ergodicity of flows defined by Rel vector field, for more general spaces $(\mathcal{L}, m_{\mathcal{L}})$, where $\mathcal{L} \subset \mathcal{H}$ is an orbit-closure for the action of $G = \mathrm{SL}_2(\mathbb{R})$ (i.e., an affine inv…
We show that the absolute period foliation of the principal stratum of abelian differentials on a surface of genus at least 3 is ergodic. We also investigate the absolute period foliation of affine invariant manifolds.
We consider diffeomorphisms in the $C^\infty$-closure of the conjugancy class of translations of the 2-torus. By a theorem of Fathi and Herman, a generic diffeomorphism in that space is minimal and uniquely ergodic. We define a new mixing-like property, which takes into account the "directions" of mixing, and we prove that generic elements of the space in question satisfy this property. As a consequence, we show that there is a residual set of strictly ergodic diffeomorphisms without invariant foliations of any kind. We also obtain an analytic version of these results.
A translation surface on (S, Σ) gives rise to two transverse measured foliations \FF, \GG on S with singularities in Σ, and by integration, to a pair of cohomology classes [\FF], \, [\GG] \in H^1(S, Σ; \R). Given a measured foliation \FF, we characterize the set of cohomology classes \B for which there is a measured foliation \GG as above with \B = [\GG]. This extends previous results of Thurston and Sullivan. We apply this to two problems: unique ergodicity of interval exchanges and flows on the moduli space of translation surfaces. For a fixed permutation σ\in \mathcal{S}_d, the space \R^d_+…
An asymptotic expansion is established for time averages of translation flows on flat surfaces. This result, which extends earlier work of A.Zorich and G.Forni, yields limit theorems for translation flows. The argument, close in spirit to that of G.Forni, uses the approximation of ergodic integrals by holonomy-invariant Hoelder cocycles on trajectories of the flows. The space of holonomy-invariant Hoelder cocycles is finite-dimensional, and is given by an explicit construction. First, a symbolic representation for a uniquely ergodic translation flow is obtained following S.Ito and A.M. Vershik…
At the beggining of the 80's, H.Masur and W.Veech started the study of generic properties of interval exchange transformations proving that almost every such transformation is uniquely ergodic. About the same time, S.Novikov's school and French mathematicians independently discovered very intriguing phenomena for classes of measured foliations on surfaces and respective IETs. For instance, minimality is exceptional in these families. A precise version of this statement is a conjecture by Novikov. The French and Russian constructions are very different ones. Nevertheless, in the most simple sit…
We analyze the rel leaf of the Arnoux-Yoccoz translation surface in genus 3. We show that the leaf is dense in the stratum $\mathcal{H}(2,2)^{\mathrm{odd}}$ but that the real-rel trajectory of the surface is divergent. On this real-rel trajectory, the vertical foliation of one surface is invariant under a pseudo-Anosov map (and in particular is uniquely ergodic), but the vertical foliations on all other surfaces are completely periodic.
We prove a substantial part of a conjecture of Kontsevich and Zorich on the Lyapunov exponents of the Teichmuller geodesic flow on the deviation of ergodic averages for generic conservative flows on higher genus surfaces. The result on the Teichmuller flow is formulated in terms of a (symplectic) cocycle on the real cohomology bundle over the moduli space of holomorphic differentials introduced by Kontsevich and Zorich. We prove that such a cocycle is non-uniformly hyperbolic, that is, all of its Lyapunov exponents are different from zero. In particular, the number of strictly positive exponen…
Topology of Foliations of the Riemann Surfaces given by the real part of generic holomorphic 1-forms, is studied. Our approach is based on the notion of Transversal Canonical Basis of Cycles (TCB) instead of using just one closed transversal curve as in the classical approach of the ergodic theory. In some cases the TCB approach allows us to present a convenient combinatorial model of the whole topology of the flow, especially effective for g=2. A maximal abelian covering over the Riemann Surface provided by the Abel Map, plays a key role in this work. The behavior of our system in the Fundame…
