Posted by: matheuscmss | November 15, 2014

## On the ergodicity of billiards in non-rational polygons

A couple of days ago (on November 12th, 2014 to be more precise), Giovanni Forni gave a talk at the “flat seminar / séminaire plat” on the ergodicity of billiards on non-rational polygons, and, by following the suggestion of two friends, I will transcript in this post my notes from Giovanni’s talk.

[Update (November 20, 2014): Some phrases near the statement of Theorem 3 below were edited to correct an inaccuracy pointed out to me by Giovanni.]

Let ${\mathcal{P}\subset \mathbb{R}^2}$ be a polygon with ${d+1}$ sides and denote by ${\theta_1, \dots, \theta_{d+1}}$ its interior angles.

The billiard flow associated to ${\mathcal{P}}$ is the following dynamical system. A point-particle in ${\mathcal{P}}$ follows a linear trajectory with unit speed until it hits the boundary of ${\mathcal{P}}$. At such an instant, the point-particle is reflected by the boundary of ${\mathcal{P}}$ (according to the usual laws of a specular reflection) and then it follows a new linear trajectory with unit speed. (Of course, this definition makes no sense at the corners of ${\mathcal{P}}$, and, for this reason, we leave the billiard flow undefined at any orbit going straight into a corner)

The phase space of the billiard flow is naturally identified with the three-dimensional manifold ${\mathcal{P}\times S^1}$: indeed, we need an element of ${\mathcal{P}}$ to describe the position of the particle and an element of the unit circle ${S^1\subset \mathbb{R}^2}$ to describe the velocity vector of the particle.

Alternatively, the billiard flow associated to ${\mathcal{P}}$ can be interpreted as the geodesic flow on a sphere ${S^2}$ with a flat metric and ${(d+1)}$ conical singularities (whose cone angles are ${2\theta_1, \dots, 2\theta_{d+1}}$) with non-trivial holonomy (see Section 2 of Zorich’s survey): roughly speaking, one obtains this flat sphere with conical singularities by taking two copies of ${\mathcal{P}}$ (one on the top of the other), gluing them along the boundaries, and by thinking of a billiard flow trajectory on ${\mathcal{P}}$ as a straight line path going from one copy of ${\mathcal{P}}$ to the other at each reflection.

This interpretation shows us that billiard flows on polygons are a particular case of geodesic flows ${\{G_t\}}$ on the unit tangent bundle ${S(M-\Sigma)}$ of compact flat surfaces ${M}$ whose subsets ${\Sigma}$ of conical singularities were removed.

Remark 1 In the case of a rational polygon ${\mathcal{P}}$ (i.e., ${\theta_1, \dots, \theta_{d+1}}$ are rational multiples of ${\pi}$), it is often a better idea (see this survey of Masur and Tabachnikov) to take several copies of ${\mathcal{P}}$ obtained by applying the finite group generated by the reflections through the sides of ${\mathcal{P}}$ and then glue by translation the pairs of parallel sides of the resulting figure. In this way, one obtains that the billiard flow associated to ${\mathcal{P}}$ is equivalent to translation (straightline) flow on a translation surface (an object that has trivial holonomy and, hence, is more well-behaved that a flat metric on ${S^2}$ with conical singularities) and this partly explains why the Ergodic Theory of billiards on rational polygons is well-developed. However, let us not insist on this point here because in what follows we will be mostly interested in billiard flows on irrational polygons.

A basic problem concerning the dynamics of billiards flows on polygons, or, more generally, geodesic flows on flat surfaces with conical singularities is to determine whether such a dynamical system is ergodic.

In view of Remark 1, we can safely skip the case of rational polygons: indeed, this setting one can use the relationship to translation surfaces to give a satisfactory answer to this problem (see the survey of Masur and Tabachnikov for more explanations). So, from now on, we will focus on billiard flows associated to non-rational polygons.

Kerckhoff, Masur and Smillie proved in 1986 that the billiard flow is ergodic for a ${G_{\delta}}$-dense subset of polygons. Their idea is to consider the ${G_{\delta}}$-dense subset of “Liouville polygons” admitting fast approximations by rational polygons (i.e., the subset of polygons whose interior angles admit fast approximations by rational multiples of ${\pi}$). Because the ergodicity of the billiard flow on rational polygons is well-understood, one can hope to “transfer” this information from rational polygons to any “Liouville polygon”.

Remark 2 The ${G_{\delta}}$-dense subset of polygons constructed by Kerckhoff, Masur and Smillie has zero measure: indeed, this happens because they require the angles ${\theta_1,\dots, \theta_{d+1}}$ to be “Liouville” (i.e., admit fast approximations by rational multiples of ${\pi}$), and, as it is well-known, the subset of Liouville numbers has zero Lebesgue measure.

A curious feature of the argument of Kerckhoff, Masur and Smillie is that it is hard to extract any sort of quantitative criterion. More precisely, it is difficult to quantify how fast the quantities ${\theta_1/\pi, \dots, \theta_{d+1}/\pi}$ must be approximated by rationals in order to ensure that the ergodicity of the billiard flow on the corresponding polygon. This happens because the genera of translation surfaces associated to the rational polygons approximating ${\theta_1,\dots,\theta_{d+1}}$ usually tend to infinity and it is a non-trivial problem to control the ergodic properties of translation flows on families of translation surfaces whose genera tend to infinity.

Nevertheless, Vorobets obtained in 1997 (by other methods) a quantitative version of Kerckhoff, Masur and Smillie by showing the ergodicity of the billiard flow on a polygon ${\mathcal{P}}$ whose interior angles ${\theta_1,\dots,\theta_{d+1}}$ verify the following fast approximation property: there exist arbitrarily large natural numbers ${N\in \mathbb{N}}$ such that

$\displaystyle \left|\theta_{1}-\pi \frac{p_1}{q_1}\right|<\left(2^{2^{2^{2^N}}}\right)^{-1}, \dots, \left|\theta_{d+1}-\pi \frac{p_{d+1}}{q_{d+1}}\right|<\left(2^{2^{2^{2^N}}}\right)^{-1}$

for some rational numbers ${p_i/q_i\in\mathbb{Q}}$, ${i=1,\dots, d+1}$, with denominators ${q_i\leq N}$, ${i=1,\dots, N}$.

In summary, the works of Kerckhoff-Masur-Smillie and Vorobets allows to solve the problem of ergodicity of the billiard flow on Liouville polygons.

Of course, this scenario motivates the question of ergodicity of billiard flows on Diophantine polygons (i.e., the “complement” of Liouville polygons consisting of those ${\mathcal{P}}$ which are badly approximated by rational polygons).

In his talk, Giovanni announced a new criterion for the ergodicity of the billiard flow on polygons (and, more generally, the geodesic flow on a flat surface with conical singularities) with potential applications to a whole class (of full measure) of Diophantine polygons.

Before stating Giovanni’s results, let us introduce some notation. Consider ${M}$ a flat surface with a finite subset ${\Sigma}$ of conical singularities (e.g., ${M\simeq S^2}$ obtained by reinterpretation of the billiard flow on a polygon). The infinitesimal structure of the unit tangent bundle ${S(M-\Sigma)}$ is described by ${3}$ vector fields:

• ${X}$ is the generator of the geodesic flow;
• ${Y}$ is the “perpendicular geodesic flow”;
• ${\Theta}$ is the generator of the rotation on the circle fibers of ${SM}$.

These vector fields satisfy the following commutation relations:

• ${[X,Y]=0}$ (because ${M}$ is a flat surface, and, hence, ${M-\Sigma}$ has zero curvature);
• ${[\Theta, X] = Y}$;
• ${[\Theta, Y]=-X}$.

Note that the knowledge of ${X, Y, \Theta}$ allows us to recover the natural Riemannian metric ${R_0}$ on ${S(M-\Sigma)}$ induced by the flat structure on ${M}$: indeed, ${R_0}$ is completely determined by the fact that ${X, Y,\Theta}$ is an orthonormal frame.

By analogy with the case of rational polygons (see this survey of Masur), we would like to apply renormalization methods to get an ergodicity criterion for the geodesic flow on ${M}$ based on the properties of the renormalization dynamics.

Logically, a naive implementation of this idea does not work: the Teichmüller geodesic flow on the moduli space of flat surfaces with arbitrary conical singularities has poor dynamical behavior (in comparison with the case of rational polygons) because these moduli spaces are usually very big and, for example, this is a serious obstruction to any recurrence property of the corresponding Teichmüller flow (which is a key ingredient in the so-called Masur’s ergodicity criterion).

Nevertheless, Giovanni noticed that one can still implement this renormalization method by introducing the following deformations of ${X, Y, \Theta}$ (playing the role of “fake Teichmüller geodesic flow”):

• ${X_s=e^{s} X}$;
• ${Y_s=e^{-s} Y}$;
• ${\Theta_s = e^{-2s}\Theta}$,

for ${s\in\mathbb{R}}$. By declaring that the vector fields ${X_s, Y_s, \Theta_s}$ form an orthonormal frame, we obtain a Riemannian metric ${R_s}$ on ${S(M-\Sigma)}$.

Remark 3 Note that ${X_s}$, ${Y_s}$ and ${\Theta_s}$ satisfy the following commutation relations:

$\displaystyle [X_s,Y_s] = [X,Y] = 0, \quad [\Theta_s, X_s] = e^{-s}[\Theta, X] = Y_s, \quad [\Theta_s, Y_s] = e^{-3s}[\Theta, Y] = -e^{-4s} X_s$

Furthermore, the volume of ${R_s}$ is ${\textrm{vol}(R_s) = e^{2s}\textrm{vol}(R_0)}$. In particular, as ${s\rightarrow\infty}$, we see that ${\textrm{vol}(R_s)\rightarrow\infty}$ and ${[\Theta_s, Y_s]\rightarrow 0}$, i.e., ${R_s}$ is very close to a Heisenberg group as ${s\rightarrow\infty}$ (i.e., its geometry becomes nilpotent in the limit). In particular, we see that the deformations ${R_s}$ of ${R_0}$ do not exhibit any sort of recurrence property (in whatever moduli space they live).

Remark 4 In the definition of ${X_s}$, resp. ${Y_s}$, the scaling factors of ${e^s}$, resp. ${e^{-s}}$, for ${X}$, resp. ${Y}$ are motivated by direct analogy with the Teichmüller geodesic flow. On the other hand, the scaling factor ${e^{-2s}}$ for ${\Theta}$ is more subtle to explain: Giovanni said that he found this scaling (which is convenient for his ergodicity criterion of billiards on polygons) from an analytical argument (see Remark 9 below). Also, Giovanni observed that, a posteriori, this scaling is “justified” from the dynamical point of view because the orbits of the geodesic flow of ${R_0}$ stay fairly close (i.e., they do not “diverge”) after applying the deformation ${R_s}$, and, in particular, one has nice “rectangles” of heights ${e^{-s}}$ and width ${e^{s}}$ (and, as it turns out, the presence of such nice rectangles is an important ingredient in Masur’s ergodicity criterion for rational polygons). However, he insisted that this “dynamical justification” was not the initial motivation to define ${\Theta_s}$ (but rather the arguments from Analysis sketched below).

In this setting, Giovanni’s ergodicity criterion for geodesic flows on flat surfaces (such as billiard flows on polygons) is:

Theorem 1 (Forni) Let ${M}$ be a flat surface with a finite subset ${\Sigma}$ of conical singularities. Suppose that there exist a subset ${P\subset\mathbb{R}}$ with positive lower density (i.e., ${\liminf\limits_{s\rightarrow+\infty}\textrm{Leb}(P\cap[-s,s])/2s > 0}$) and a real number ${\varepsilon_0>0}$ such that for each ${s\in P}$ and ${0<\varepsilon<\varepsilon_0}$ one can find a connected subset ${\Omega_s(\varepsilon)\subset SM}$ with the following properties:

• (i) ${\limsup\limits_{\substack{s\in P \\ s\rightarrow \infty}} h_s(\Omega_s(\varepsilon))>0}$ for all ${0<\varepsilon<\varepsilon_0}$, where ${h_s(\Omega_s(\varepsilon))}$ denotes the Cheeger constant of ${\Omega_s(\varepsilon)}$ with respect to ${R_s}$ (see below for the definitions);
• (ii) ${\lim\limits_{\varepsilon\rightarrow 0} \textrm{vol}_{R_0}(\Omega_s(\varepsilon))=1}$ uniformly on ${s\in P}$.

Then, the geodesic flow ${\{G^t\}}$ on ${M}$ is ergodic.

Remark 5 Recall that the Cheeger constant ${h_R(\Omega)}$ of a domain ${\Omega\subset SM}$ with respect to a Riemannian metric ${R}$ on ${SM}$ is

$\displaystyle h_R(\Omega) = \inf\limits_{\mathcal{S} \textrm{ separating surface }}\frac{\textrm{Area}(\mathcal{S}\cap \Omega)}{\min\{\textrm{vol}(\Omega_{\mathcal{S}}'), \textrm{vol}(\Omega_{\mathcal{S}}'')\}}$

where ${\Omega_{\mathcal{S}}'}$ and ${\Omega_{\mathcal{S}}''}$ are the connected components of ${\Omega-\mathcal{S}}$.

Intuitively, Giovanni’s ergodicity criterion can be thought as saying that if we can find a suitable subset ${P}$ of good renormalization times in the sense that the complement ${\Omega_s(\varepsilon)}$ of “adequate small neighborhoods” of the subset ${\Sigma}$ of conical singularities has bounded geometry (i.e., a controlled Cheeger constant, cf. the condition (i) above) and almost full volume (cf. the condition (ii) above), then we can exploit these renormalization times to conclude the ergodicity of the geodesic flow.

Remark 6 For the sake of comparison with the case of rational polygons/translation surfaces, let us observe that for a translation surface ${M}$ (with flat metric ${R}$) one has

$\displaystyle h_R(M)\geq c\cdot\textrm{sys}(M)$

where ${c}$ is a constant depending only on the genus of ${g}$ and ${\textrm{sys}(M)}$ denotes the systole of ${M}$ (that is, the length of the shortest saddle connection). In particular, since the systole of a translation surface on a compact region of the moduli space admits an uniform lower bound, the analog of the condition (i) in Giovanni’s ergodicity criterion in the setting of translation surfaces is satisfied by most translation surfaces thanks to the recurrence properties of the Teichmüller geodesic flow (that is, of the deformation ${X_s=e^{s}X}$, ${Y_s=e^{-s}Y}$ and ${\Theta_s=\Theta}$).

Remark 7 Still for the sake of comparison, it is worth to observe that after more recent works of Cheung-Eskin and Treviño we know that the ergodicity criterion can be substantially improved in the context of translation surfaces: indeed, one can ensure the ergodicity (and even unique ergodicity) of the flow generated by ${X}$ whenever the systole ${\delta(s)}$ of the flat metric ${R_s}$ associated to the Teichmüller deformation ${X_s=e^sX}$, ${Y_s=e^{-s}Y}$ (and ${\Theta_s=\Theta}$) verifies the non-integrability condition

$\displaystyle \int_0^{\infty}\delta(s)^2 ds = \infty$

(Note that this non-integrability condition is automatic for recurrent Teichmüller deformations as for such deformations the quantity ${\delta(s)}$ admit uniform lower bounds on a countable family of disjoint subintervals of definite sizes) Evidently, these results of Cheung-Eskin and Treviño motivate the following question: is it possible to weaken the condition (i) in Theorem 1 in order to allow Cheeger constants ${h_s(\Omega_s(\varepsilon))}$ that could approach ${0}$ slowly (maybe in a similar spirit of the non-integrability condition above)? In fact I asked this question to Giovanni after his talk and he pointed out that it is not very clear that this possible with his current argument because of the subtle nature of the proof of the estimate (1) appearing below (especially the estimate of the term ${e^{-4s}\|\Theta \textrm{Re}(h_s^{\pm})\|_{L^2}^2}$).

Before discussing some elements of the proof of Theorem 1, let us quickly comment on the potential applications of Giovanni’s ergodicity criterion. At first sight, it is not obvious at all how to decide whether a given polygon ${\mathcal{P}}$ with interior angles ${\theta_1, \dots, \theta_{d+1}}$ (or, more generally, a flat surface ${M}$ with conical singularities with cone angles ${\theta_1,\dots, \theta_{d+1}}$) verify the requirements of Theorem 1 (especially the condition (i)).

In this direction, even though Giovanni said that he has not fully checked his arguments yet, Giovanni is confident that the following Diophantine conditions on ${\theta_1,\dots,\theta_{d+1}}$ are sufficient to apply his ergodicity criterion.

Theorem 2 (Forni (in progress)) Let ${\mathcal{P}}$ be a polygon with ${d+1}$ sides and interior angles ${\theta_1,\dots,\theta_{d+1}}$.Denote by ${\overline{\theta}=(\theta_1,\dots,\theta_d)}$ (see Remark 8 below for the reason why we exclude ${\theta_{d+1}}$). Suppose that ${\overline{\theta}}$ satisfies the following Diophantine conditions:

• (1) there exists a constant ${C}$ such that ${\sum\limits_{i=1}^d \|N\frac{\theta_i}{2\pi}\|_{\mathbb{Z}}\geq \frac{C}{\sqrt{N}}}$ for all ${N\in\mathbb{N}}$
• (2) there exists a constant ${C}$ such that for all (non-trivial) integer vectors ${v_1,\dots, v_{d-1}\in \mathbb{Z}^d-\{0\}}$ one has

$\displaystyle |v_1\wedge\dots\wedge v_{d-1}|^2 \left(\sum\limits_{i=1}^{d-1} |v_1|\dots|v_{i-1}|\cdot\| \frac{\langle v_i, \overline{\theta}\rangle}{2\pi}\|_{\mathbb{Z}} \cdot|v_{i+1}|\dots|v_{d-1}|\right)\geq C$

Then, the conditions (i) and (ii) in Theorem 1 hold, and, a fortiori, the billiard flow on ${\mathcal{P}}$ is ergodic.

Remark 8 The sum ${\sum\limits_{i=1}^{d+1}\theta_i = (d-1)\pi}$ of the interior angles of ${\mathcal{P}}$ is a fixed rational multiple of ${\pi}$. For this reason, it is natural to impose Diophantine conditions on ${(\theta_1,\dots, \theta_d)}$ rather than ${(\theta_1,\dots, \theta_{d+1})}$.

Even though we are not going to sketch the proof of Theorem 2 today, let us now make two comments on the Diophantine conditions (1) and (2).

First, these conditions do not seem totally independent (even though it is not easy to figure out their relationship): for example, for ${d=2}$, the condition (2) becomes ${|v_1|^2\|\frac{\langle v_1,\overline{\theta}\rangle}{2\pi}\|_{\mathbb{Z}}\geq C}$, that is, ${\|\frac{\theta_1}{2\pi} n_1 + \frac{\theta_2}{2\pi} n_2\|_{\mathbb{Z}}\geq \frac{C}{n_1^2+n_2^2}}$ for all ${v_1=(n_1,n_2)\in\mathbb{Z}^2-\{(0,0)\}}$, and this latter condition resembles the condition (1).

Secondly, the condition (1) is a full Lebesgue measure condition on ${\overline{\theta}=(\theta_1,\dots,\theta_d)}$ only for ${d\geq 3}$. In other terms, one can use Theorem 2 to deduce the ergodicity of the billiard flow on almost every polygon ${\mathcal{P}}$ with ${d+1\geq 4}$ sides, but the analogous statement for the case of triangles remains still open.

Closing this post, let us give a brief sketch of the proof of Giovanni’s ergodicity criterion (Theorem 1).

The argument starts in the same way as in Giovanni’s proof of the spectral gap property (“${\lambda_2<1=\lambda_1}$”) for the Lyapunov exponents of the Kontsevich-Zorich cocycle via variational formulas for the Hodge norm (in Section 2 of this paper here). More concretely, we consider the foliated Cauchy-Riemann operators

$\displaystyle \partial_s^{\pm}:= X_s \pm i Y_s$

associated to the deformation ${(X_s,Y_s,\Theta_s)}$. (We said “foliated” because the distribution ${\{X_s,Y_s\}}$ is integrable in ${SM}$ and ${\partial_s^{\pm}}$ are the usual ${\partial}$ and ${\overline{\partial}}$ along the leaves of this foliation)

Next, given a ${L^2}$-function ${u}$, we consider its decomposition

$\displaystyle u=\partial_s^+ v_s^+ + h_s^- = \partial_s^- v_s^- + h_s^+$

in terms of the image and the kernel of the Cauchy-Riemann operators ${\partial_s^{\pm}}$. (Here, there is a subtle point: contrary to the case of translation surfaces, it is not known that the image of ${\partial_s^{\pm}}$ is closed; in particular, one should replace ${\partial_s^+ v_s^+}$ and ${\partial_s^- v_s^-}$ by adequate elements in the closure of the images of ${\partial_s^{\pm}}$, but we will skip this technical detail by pretending that the decomposition above can always be made)

Recall that, under the assumptions of Theorem 1, our task is to show that the geodesic flow is ergodic, that is, we want to show that any real ${L^2}$-function ${u}$ with ${Xu=0}$ (i.e., ${u}$ is invariant) is actually constant.

For this sake, by mimicking the proof of Lemma 2.1′ of his paper, Giovanni shows the following variational formula:

$\displaystyle \frac{d}{ds}\|h_s^{\pm}\|_{L^2(SM)}^2 = 4\|\textrm{Im } h_s^{\pm}\|^2_{L^2(SM)}$

From this formula, we can deduce that ${\|\textrm{Im } h_s^{\pm}\|_{L^2(SM)}\rightarrow 0}$ as ${s\rightarrow\infty}$, ${s\in P}$ (where ${P\subset\mathbb{R}}$ is the subset of positive lower density of “good renormalization times”, cf. the statement of Theorem 1). Indeed, since ${h_s^{\pm}}$ is obtained by orthogonal projection of ${u}$ with respect to the (closure of the) image of ${\partial_s^{\pm}}$, we have that ${\|h_s^{\pm}\|_{L^2}\leq \|u\|_{L^2}}$ is uniformly bounded for all ${s\in\mathbb{R}}$. By plugging this information into the variational formula above, we obtain that

$\displaystyle 4\int_0^S \|\textrm{Im }h_s^{\pm}\|_{L^2}^2 = \|h_S^{\pm}\|_{L^2}^2 - \|h_0\|_{L^2}^2\leq 2\|u\|_{L^2}^2$

for all ${S\in\mathbb{R}}$ and the claim that ${\|\textrm{Im } h_s^{\pm}\|_{L^2(SM)}\rightarrow 0}$ as ${s\rightarrow\infty}$, ${s\in P}$ follows.

In other terms, we have just shown that ${\textrm{Re } h_s^{\pm}}$ converges (in ${L^2}$) to ${u}$ as ${s\rightarrow\infty}$, ${s\in P}$.

Next, we observe that the functions ${\textrm{Re }h_s^{\pm}}$ are harmonic (since ${h_s^{\pm}}$ are meromorphic, resp. anti-meromorphic), and, thus, we can apply Cauchy’s estimate to obtain that

$\displaystyle \|\nabla_s\textrm{Re }h_s^{\pm}\|_{L^2(SM-V_{\Sigma}(\varepsilon))}\leq \frac{C}{\varepsilon} \|\textrm{Im } h_s^{\pm}\|_{L^2}$

where ${\nabla_s}$ is the gradient in the metric ${R_s}$ associated to the deformation ${(X_s, Y_s, \Theta_s)}$, and ${V_{\Sigma}(\varepsilon)}$ is a ${\varepsilon}$-neighborhood of ${\Sigma}$ in ${SM}$ (that is, ${SM-V_{\Sigma}(\varepsilon)}$ is essentially equal to the subset ${\Omega_s(\varepsilon)}$ by condition (ii) of Theorem 1).

Using the facts that ${\nabla_s}$ has “bounded geometry” (by condition (i) of Theorem 1), ${\|\nabla_s\textrm{Re }h_s^{\pm}\|_{L^2(SM-V_{\Sigma}(\varepsilon))}\leq \frac{C}{\varepsilon}\|\textrm{Im } h_s^{\pm}\|_{L^2}\rightarrow 0}$ and ${\textrm{Re } h_s^{\pm}\rightarrow u}$ as ${s\rightarrow\infty}$, ${s\in P}$, we (get that ${u}$ is constant along the leaves of the foliation associated to ${\{X_s, Y_s\}}$) see that one is getting closer to show that ${u}$ is constant.

Nevertheless, the information obtained in the previous paragraph is not quite sufficient to conclude that ${u}$ is constant because the leaves of the foliation associated to ${\{X_s, Y_s\}}$ (sometimes called Loch Ness monsters in the flat surfaces literature, see, e.g, this paper here) might not have bounded geometry. For this reason, Giovanni needs also At this point, it remains only to control the behavior of ${u}$ in the ${\Theta}$-direction. Here, after replacing ${\textrm{Re } h_s^{\pm}}$ by an adequate truncation of its Fourier series in the ${\Theta}$-direction still called ${\textrm{Re }h_s^{\pm}}$ by a slight abuse of notation, Giovanni told us (without giving the proof because he ran out of time) that a computation based on arguments from Harmonic Analysis reveals that

Theorem 3 One has

$\displaystyle \|\nabla_s\textrm{Re }h_s^{\pm}\|_{L^2(SM-V_{\Sigma}(\varepsilon))}^2 + e^{-4s}\|\Theta \textrm{Re } h_s^{\pm}\|_{L^2}^2 \rightarrow 0 \ \ \ \ \ (1)$

as ${s\rightarrow \infty}$, ${s\in P}$.

Because ${\Theta_s:=e^{-2s}\Theta}$, the bounded geometry condition (i) in Theorem 1 allows us to conclude that (${u}$ is also constant along the ${\Theta}$-direction. Therefore, we deduce that) ${u}$ is constant on ${SM}$, and, hence the geodesic flow (generated by ${X}$) is ergodic (so that the sketch of proof of Theorem 1 is complete).

Remark 9 As we mentioned in Remark 4 above, Giovanni’s choice of deformation ${\Theta_s=e^{-2s}\Theta}$ in the ${\Theta}$-direction was purely guided by the arguments from Harmonic Analysis in the proof of Theorem 3 which “impose” the factor of ${e^{-4s}}$ in his control of the growth of ${\|\Theta \textrm{Re } h_s^{\pm}\|_{L^2}^2}$.