Posted by: matheuscmss | March 5, 2013

Eskin-Kontsevich-Zorich regularity conjecture III: accessing deep levels of the systole function

The plan of today’s discussion is to use the tools (“orbit by orbit estimates” and a variant of Rokhlin disintegration theorem) from the previous post to study the following question (stated after the picture).

Let {X=\{M\in \mathcal{C}: \textrm{sys}(M)=\rho\}} be a certain (fixed) level of the systole function on a connected component {\mathcal{C}} of a stratum of the moduli space of translation surfaces of genus {g\geq2}. Denote by {X_0^*} the set of {M\in X} such that all non-vertical saddle-connections have length {>\rho} and, for each {T>0}, consider the set {Y^*(T)} of translation surfaces {M} with systole {\textrm{sys}(M)<\rho\exp(-T)} having the form {M=g_t R_{\theta} M_0} for some {M_0\in X}, {|\theta|<\pi/4} and {0<t<\log\cot|\theta|}. In other words, using the notation {J(T,\theta):=\{t\in\mathbb{R}:\|g_t R_{\theta} e_2\|<\exp(-T)\}} introduced in the previous post,

\displaystyle Y^*(T):=\{M=g_t R_{\theta} M_0: M_0\in X_0^*, |\theta|<\pi/4, t\in J(T,\theta)\}

Geometrically, {Y^*(T)} consists of the pieces of arcs of hyperbola below the threshold {\rho\exp(-T)} in the figure below:


In this notation, given a {SL(2,\mathbb{R})}-invariant probability measure {m} on {\mathcal{C}}, we want to compute the {m-}measure of {Y^*(T)} in terms of {X_0^*}, that is, we want to determine how “fat” is the set {Y^*(T)} of translation surfaces that are “accessible” (via {g_t} and {R_{\theta}} movements) from {X_0^*}.

In fact, a precise answer to this question will occupy this entire post and, in the next (and last) post of this series, we will use this answer to estimate the {m}-measure of the set {\mathcal{C}_2(\rho\exp(-T))} (of translation surfaces with two non-parallel saddle-connections of lengths {\leq\rho\exp(-T)}) as follows. Firstly, we will see that {Y^*(T)} captures “almost all” translation surfaces in {\{M\in\mathcal{C}: \textrm{sys}(M)\leq\rho\exp(-T)\}} (for {\rho}, i.e., {X_0^*}, conveniently chosen). In particular, we will reduce the problem of measuring {\mathcal{C}_2(\rho\exp(-T))} to the task of estimating {\mathcal{C}_2(\rho\exp(-T))\cap Y^*(T)}. Then, we will see that the translation surfaces {M_0\in X_0^*} generating a translation surface {M=g_t R_{\theta} M_0\in \mathcal{C}_2(\rho\exp(-T))\cap Y^*(T)} have a pair of saddle-connections with a very small angle {\leq\theta_0} (with {\theta_0} tending to zero as {T\rightarrow\infty}) and it is not hard to see that this angle condition corresponds to a subset of {X_0^*} with arbitrarily small “density” in {X_0^*}.

We organize this post into two sections. In the first section we will explain how the variant of Rokhlin disintegration theorem from the previous post of this series can be used to describe the {m}-measure of {Y^*(T)} in terms of the measure of {X_0^*} with respect to a certain {m_0} obtained after disintegrating {m} along certain pieces of {SL(2,\mathbb{R})}-orbits. Then, in the second section, we will explain why the measure {m_0} obtained from this disintegration process is a sort of “density” measure.

1. Construction of the “density measure” {m_0} on {X_0^*}

Let {m} be a {SL(2,\mathbb{R})}-invariant probability measure on a connected component of a stratum of the moduli space of translation surfaces of genus {g\geq 2}. Given {\rho>0} such that {m(\{M\in\mathcal{C}:\textrm{sys}(M)>\rho\})>0}, consider the set

\displaystyle Y^*:=\{M=g_t R_{\theta} M_0: M_0\in X_0^*, |\theta|<\pi/4 \textrm{ and } 0<t<\log\cot|\theta|\}

of translation surfaces with systole {<\rho} accessible from {X_0^*} from {g_t} and {R_{\theta}} movements, where {X_0^*} is the set of translation surfaces with systole {\rho} such that all non-vertical saddle-connections have length {>\rho}.

Note that we can write

\displaystyle Y^*=\bigsqcup\limits_{|\theta|<\pi/4}\bigsqcup\limits_{0<t<\log\cot|\theta|} g_t R_{\theta} (X_0^*)

where the symbol {\bigsqcup} denotes a disjoint union: indeed, if {g_t R_{\theta} M_0=g_{t'} R_{\theta'} M_0'} with {t\geq t'}, then {R_{\theta'} M_0'=g_{t-t'} R_{\theta} M_0}; since {R_{\theta'}M_0'} and {R_{\theta} M_0} have systole {\rho} and {0\leq t-t'<\log\cot|\theta|}, it follows that {t=t'}; in particular, {R_{\theta'} M_0' = R_{\theta} M_0} and, from the definition of {X_0^*}, we also deduce that {\theta=\theta'}.

Next, we observe that the infinitesimal generator of {n_u=\left(\begin{array}{cc} 1 & 0 \\ u & 1\end{array}\right)} is tangent to {X_0^*}. Indeed, since {n_u} fixes the vertical direction while changing the horizontal direction only a little bit if {u} is close to zero, we have that {n_u M_0\in X_0^*} for all {M_0\in X_0^*} and {u} sufficiently close to zero.

Remark 1 Here, we implicitly used that {X_0^*} is a manifold when talking about tangent vectors to {X_0^*}. While this is true if the translation surfaces in {\mathcal{C}} have no non-trivial automorphisms (symmetries), in general {X_0^*} is an orbifold but not a manifold. However, this little technical problem is easy to overcome: one can either take finite covers of {X_0^*} that are manifolds (e.g., one can mark horizontal separatrices of the translation surfaces to “kill” automorphisms), or one can write {\mathcal{C}} as a finite union of {SL(2,\mathbb{R})}-invariant manifolds (as in our preprint with Artur and Jean-Christophe). In any event, for the sake of this post, we will pretend that {\mathcal{C}} is a manifold.

In particular, since {X_0^*} is a codimension {2} submanifold of {\mathcal{C}} (we are fixing the value of the systole, a codimension 1 condition, and the direction of the lenght-minimizing saddle-connections, another codimension 1 condition), we can select near any point {M_0\in X_0^*} a codimension {3} submanifold {M_0\in\Sigma\subset X_0^*} locally transverse to the infinitesimal generator of {n_u}. Using these {\Sigma}‘s, we can represent {Y^*} as

\displaystyle g_tR_{\theta} n_u M

with {M\in\Sigma}, {u} close to {0}, {|\theta|<\pi/4} and {0<t<\log\cot|\theta|}. At this point, we are ready to use the technology of the previous post to understand the measure {m} on {Y^*} in terms of a density measure {m_0} on {X_0^*}. Indeed, we begin by noticing that {Y^*} has positive {m}-measure. In fact, by Fubini’s theorem and the {SL(2,\mathbb{R})}-invariance of {m}, we have that

\displaystyle m(B)=\int_{\mathcal{C}}\nu(\{g\in SL(2,\mathbb{R}): gx\in B\}) dm(x)

where {\nu} is any probability measure on {SL(2,\mathbb{R})}. By taking {\nu} a “normalized” version of Haar measure of {SL(2,\mathbb{R})} (say multiplied by an adequate everywhere positive density in order to get a probability measure) and by noticing that, for each {x\in M} with {\textrm{sys}(x)>\rho}, the set of elements {g\in SL(2,\mathbb{R})} has non-empty interior in {SL(2,\mathbb{R})} and thus positive {\nu}-measure, we deduce that {m(Y^*)} is positive as soon as the {m}-measure of the set {\{x\in\mathcal{C}: \textrm{sys}(x)>\rho\}} is positive.

Once we know that {m(Y^*)>0}, we can apply the variant of Rokhlin’s theorem in the previous post to the {m|_{Y^*}} to deduce that locally

\displaystyle m|_{Y^*}=\textrm{Haar}|_{W}\times \nu

where {W=\{g_t R_{\theta} n_u\in SL(2,\mathbb{R}): |\theta|<\pi/4, 0<t<\log\cot|\theta|, u\in\mathbb{R}\}} and {\nu} is a finite measure on {\Sigma}.

Next, we recall from the Subsection “The decomposition {g_t R_{\theta} n_u} and Haar measure on {SL(2,\mathbb{R})}” of the previous post that {\textrm{Haar}|_{W}=dt\times cos2\theta d\theta\times du}. Therefore, we can write

\displaystyle m|_{Y^*}=\cos 2\theta dt\,d\theta\, m_0 \ \ \ \ \ (1)

where {m_0} is a finite measure on {X_0^*}.

From this formula, we can answer the question at the beginning of this post, i.e., we can compute the {m}-measure of the set

\displaystyle Y^*(T)=\{M=g_t R_{\theta} M_0: M_0\in X_0^*, |\theta|<\pi/4, t\in J(T,\theta)\}

In fact, by (1)

\displaystyle m(Y^*(T))=m_0(X_0^*)\int_{-\pi/4}^{\pi/4} |J(T,\theta)|\cos 2\theta d\theta

where {J(T,\theta)=\{t\in\mathbb{R}: \|g_t R_\theta e_2\|<\exp(-T)\}} and {|J(T,\theta)|} is the length of the interval {J(T,\theta)\subset\mathbb{R}}. Now, we recall that the quantity {|J(T,\theta)|} was computed in Subsection “On the action of the diagonal subgroup {g_t=\textrm{diag}(e^t, e^{-t})}” of the previous post:

\displaystyle |J(T,\theta)|=\frac{1}{2}\log\frac{1+\cos\omega}{1-\cos\omega}

where {\sin2\theta:=\exp(-2T)\sin\omega}. By plugging this into the integral expression for {m(Y^*(T))} above, we get that:

\displaystyle m(Y^*(T))=m_0(X_0^*)\exp(-2T)\int_{-\pi/2}^{\pi/2}\left(\frac{1}{2}\log\frac{1+\cos\omega}{1-\cos\omega}\right)\left(\frac{1}{2}\cos\omega\right) d\omega

Finally, the integral on the right hand-side was computed in the Subsection “The decomposition {g_t R_{\theta} n_u} and Haar measure on {SL(2,\mathbb{R})}” of the previous post and its value is {\pi}. In summary,

Proposition 1

\displaystyle m(Y^*(T))=\frac{1}{2}\pi \, m_0(X_0^*) \exp(-2T) \ \ \ \ \ (2)

In other terms, the {m}-measure of the subset {Y(T^*)} of {\{M:\textrm{sys}(M)\leq\rho\exp(-T)\}} captured by {X_0^*} is an explicit (and simple) function of the quantity {m_0(X_0^*)} where {m_0} was obtained from {m} after disintegration along pieces of {W}-orbits.

Intuitively, we want to think of {m_0} as a sort of “density measure” on {X_0^*} and we wish to say that {X^*=\bigcup_{|\theta|<\pi/2} R_{\theta} X_0^*} occupies most of the level {X=\{M: \textrm{sys}(M)=\rho\}} of the systole function, but these properties are not automatic from Rokhlin’s disintegration theorem. On the other hand, as we are going to explain in the next section, we dispose of all elements to give a precise meaning for the intuition that {m_0} is a density measure.

2. Why is {m_0} a density measure?

The main result of this section is:

Proposition 2 For each {\tau>0}, denote by {S(\tau)} the slice

\displaystyle S(\tau)=\{M\in\mathcal{C}:\rho\geq\textrm{sys}(M)\geq\rho\exp(-T)\}


\displaystyle \lim\limits_{\tau\rightarrow0}\frac{1}{\tau}m(S(\tau))=\pi\, m_0(X_0^*)

Of course, this proposition says that {m_0} is a density measure in the sense that {m_0(X_0^*)} can be recovered from the {m}-measures of slices {S(\tau)} near the level set {X=\{M\in\mathcal{C}: \textrm{sys}(M)=\rho\}} of the systole function.

We will divide the proof of this proposition into two parts. Firstly, we will construct a regular part of the slice {S(\tau)} consisting of translation surfaces obtained by applying a “twisted Teichmüller flow” to {X_0^*} and we will show that this regular part has {m}-measure {\sim\tau\,\pi\,m_0(X_0^*)}; in particular, this computation of the {m}-measure of the regular part of {S(\tau)} shows that {\liminf\limits_{\tau\rightarrow0}\frac{1}{\tau}m(S(\tau))\geq\pi m_0(X_0^*)}. Secondly, we will show that the {m}-measure of the singular part of the slice {S(\tau)}, i.e., the complement of the regular part, is {o(\tau)} and this will complete the proof of the desired proposition.

In order to organize the discussion of Proposition 2, we will treat the regular and singular parts of {S(\tau)} separately in the following two subsections below.

2.1. Regular part of slices

Let {X^*=\bigcup R_{\theta} X_0^*} and consider the following “twisted Teichmüller flow”:

\displaystyle \Phi_t(M):=R_{\theta}g_tR_{-\theta} M

when {M\in R_{\theta}(X_0^*)}. Note that {\Phi_t} is injective and {\Phi_t(M)} has systole {\rho e^{-t}}.

In this language, the regular part {Reg(\tau)} of the slice {S(\tau)=\{\rho\geq\textrm{sys}(M)\geq\rho\exp(-\tau)\}} is:

\displaystyle Reg(\tau):=\bigsqcup\limits_{0\leq t\leq\tau}\Phi_t(X^*)

For the computation of the {m}-measure of {Reg(\tau)}, it is convenient to study the following measure on {X^*}:

\displaystyle m_{\tau}(B):=\frac{2}{1-\exp(-2\tau)}m(\bigsqcup\limits_{0\leq t\leq\tau}\Phi_t(B))

for any Borel set {B\subset X^*}.

Proposition 3 {m_{\tau}} doesn’t depend on {\tau}: in fact, {m_{\tau}=d\theta\times m_0} on {X^*=\bigsqcup\limits_{|\theta|<\pi/2} R_{\theta}(X_0^*)}.

Proof: As we saw in the previous section, {m|_{Y^*}=\cos2\theta\,dt\,d\theta\,m_0} if we write {M\in Y^*} as {M=g_t R_{\theta} M_0} with {M_0\in X_0^*}.

On the other hand, by definiton, if {M=R_{\theta}M_0} with {M_0\in X_0^*}, then

\displaystyle \Phi_t(M)=R_{\theta} g_t (M_0)

So, we will be able to compute {m_{\tau}} if we can express {R_{\theta}g_t} in terms of the {g_t R_{\theta} n_u} decomposition, i.e.,

\displaystyle R_{\theta} g_t = g_{T(t,\theta)} R_{\Theta(t,\theta)}n_{U(t,\theta)}

Note that, for {\theta} small enough, the vector {R_{\theta}g_t e_2=(b,d)} satisfies {d>0} and {|bd|<1/2}. By the results in Subsection “The decomposition {g_tR_{\theta}n_u} and the Haar measure of {SL(2,\mathbb{R})}” of the previous post, it follows that {R_{\theta}g_t} has a decomposition {g_{T(t,\theta)} R_{\Theta(t,\theta)} n_{U(t,\theta)}}. Furthermore, by a direct computation similar to the one in Subsection “The decomposition {g_tR_{\theta}n_u} and the Haar measure of {SL(2,\mathbb{R})}” of the previous post for the computation of the density of Haar measure in {g_tR_{\theta}n_u}-coordinates, one can check that, for {\theta} close to zero, {T(t,\theta)=t+O(\theta)}, {\Theta(t,\theta)=e^{-2t}\theta+O(\theta^2)} and {U(t,\theta)=O(\theta)}, so that

\displaystyle m_{\tau}(\bigsqcup\limits_{s\in[0,\theta]} R_s(B_0))=\frac{2}{1-\exp(-2\tau)}\left(\int_0^{\tau}e^{-2t}\theta dt\right)m_0(B_0)+O(\theta^2)

\displaystyle =m_0(B_0)\theta+O(\theta^2)

for any Borel set {B_0\subset X_0^*}. It follows that {m_{\tau}=d\theta\times m_0} as desired. \Box

Corollary 4 The {m}-measure of the regular part of the slice is:

\displaystyle m(Reg(\tau))=\frac{1-\exp(-2\tau)}{2}\pi \, m_0(X_0^*).

In particular,

\displaystyle \lim\limits_{\tau\rightarrow0}\frac{1}{\tau}m(Reg(\tau))=\pi\,m_0(X_0^*).

2.2. Singular part of slices

The main proposition of this subsection is:

Proposition 5 The singular part {Sing(\tau):= S(\tau) - Reg(\tau)} of the slice {S(\tau)} has {m}-measure

\displaystyle m(Sing(\tau))=o(\tau)

as {\tau\rightarrow0}.

Observe that, by definition, the singular part of {S(\tau)} is contained in the set {Z(\tau)} of translation surfaces {M\in S(\tau)} with a saddle-connection of length {\leq\rho\exp(\tau)} non-parallel to a length-minimizing one. For each {M\in Z(\tau)}, let {\theta(M)} denote the minimal angle between two non-parallel saddle-connections of length {\leq 6\rho}.

The proof of Proposition 5 follows from the following two lemmas:

Lemma 6 (“small angles”) Given {\eta>0}, there exists {\theta_0=\theta_0(\eta)>0} such that

\displaystyle m(\{M\in Z(\tau):\theta(M)<\theta_0\})<\eta\tau

Lemma 7 (“big angles”) For each {\theta_0>0},

\displaystyle m(\{M\in Z(\tau):\theta(M)\geq\theta_0\})=O_{\theta_0}(\tau^{3/2})

Let us start with the case of big angles. By Fubini’s theorem and the {SL(2,\mathbb{R})}-invariance of {m}, we have that

\displaystyle m(B)=\int_{\mathcal{C}} \frac{\textrm{Haar}(\{g\in SL(2,\mathbb{R}): \|g\|\leq 2, gx\in B\})}{\textrm{Haar}(\{g\in SL(2,\mathbb{R}):\|g\|\leq 2\})} dm(x)

for any Borel set {B\subset\mathcal{C}}, where {\textrm{Haar}} is a Haar measure on {SL(2,\mathbb{R})}. By applying this formula with {B=\{M\in Z(\tau):\theta(M)\geq\theta_0\}}, our task is reduced to estimate the quantity

\displaystyle \frac{\textrm{Haar}(\{g\in SL(2,\mathbb{R}): \|g\|\leq 2, gx\in Z(\tau), \theta(gx)\geq\theta_0\})}{\textrm{Haar}(\{g\in SL(2,\mathbb{R}):\|g\|\leq 2\})}

for each {x\in\mathcal{C}}. By definition, if {gx\in Z(\tau)} and {\theta(gx)\geq\theta_0}, then {x} has systole {\rho/3} (for {\tau} small enough) and {x} has a pair of non-parallel saddle-connections {\gamma_1} and {\gamma_2} of lengths {\leq 3\rho} and angle {\geq\theta_0/100} such that {g(\gamma_1)} and {g(\gamma_2)} have lengths between {\rho\exp(-\tau)} and {\rho\exp(\tau)}. In other words, using the notations from Subsection “Euclidean norms under {SL(2,\mathbb{R})}action on {\mathbb{R}^2}” from the previous post,

\displaystyle \{g\in SL(2,\mathbb{R}):\|g\|\leq 2, gx\in Z(\tau), \theta(gx)\geq\theta_0\}\subset \bigcup\limits_{v,v'}E(v/\rho,v'/\rho,\tau)

where {v} and {v'} are non-parallel holonomy vectors of saddle-connections of {x} with lengths between {\rho\exp(-\tau)} and {\rho\exp(\tau)} and making angle {\theta_0/100}. From the main result of the Subsection “Euclidean norms under {SL(2,\mathbb{R})}action on {\mathbb{R}^2}” from the previous post, the Haar measures of each of the sets {E(v/\rho,v'/\rho,\tau)} are {O(\tau^{3/2})} where the implied constant depend only on {\|v\pm v'\|/\rho}, i.e., {\theta_0}. Thus, the proof of Lemma 7 will be complete once we dispose of upper bounds on the number of pairs {v, v'} as above and, as a matter of fact, these bounds exist in the literature: indeed, among several other things, H. Masur (see also Theorem 5.4 of this paper of A. Eskin and H. Masur) showed that the number of saddle-connections of length {\leq 3\rho} of a translation surface {x} with systole {\geq\rho/3} is uniformly bounded by a constant depending only on {\rho}.

Now, let us deal with the case of small angles. Given {\eta>0}, we can select {\theta_1>0} small enough so that the {m}-measure of the set {N(\theta_1)} of translation surfaces with two non-parallel saddle-connections with angle {\leq\theta_1} and lengths between {\rho/6} and {6\rho} is {m(N(\theta_1))<\eta/3}. Next, we select {\theta_0>0} small enough so that

\displaystyle g_t(\{M\in Z(\tau): \theta(M)<\theta_0\})\subset N(\theta_1)

for all {0\leq t\leq 2}. In particular, if we consider the subset {Z_1(\tau)} of {Z(\tau)} consisting of {M\in Z(\tau)} such that a length-minimizing saddle-connection makes angle {\leq\pi/6} with the vertical, then, from the fact that {\theta(M)<\theta_0}, one can check that the sets {g_{j\tau}(\{M\in Z_1(\tau):\theta(M)<\theta_0\})} are mutually disjoint for {j=0, \dots, \lfloor1/\tau\rfloor}. By combining this fact with the {g_t}-invariance of {m}, we deduce that

\displaystyle \frac{1}{\tau}m(\{M\in Z_1(\tau): \theta(M)<\theta_0\})

\displaystyle \leq m\left(\bigsqcup\limits_{j=0}^{\lfloor1/\tau\rfloor}g_{j\tau}(\{M\in Z_1(\tau): \theta(M)<\theta_0\})\right)

\displaystyle \leq m(N(\theta_1))

Since {m(N(\theta_1))<\eta/3}, we conclude that

\displaystyle m(\{M\in Z_1(\tau): \theta(M)<\theta_0\})\leq\eta\tau/3

Finally, since {Z_1(\tau)} differs from {Z(\tau)} only by the extra condition that a length-minimizing saddle-connection makes angle {\leq\pi/6} with the vertical, we can use the {R_\theta}-invariance of {m} to obtain that

\displaystyle m(\{M\in Z(\tau):\theta(M)<\theta_0\})\leq 3m(\{M\in Z_1(\tau):\theta(M)<\theta_0\})\leq \eta\tau.

Of course, this completes the proof of Lemma 6.

At this point, we can summarize the discussion of this post as follows. We wrote {m|_{Y^*}} as {\cos2\theta dt\,d\theta m_0} and we used this to show that the {m}-measure of the set {Y^*(T)} of translation surfaces accessible from {X_0^*} by {g_t} and {R_{\theta}} movements is

\displaystyle m(Y^*(T))=\pi m_0(X_0^*) \exp(-2T)

where {m_0} is a “density” measure in the sense that

\displaystyle \lim\limits_{\tau\rightarrow0}\frac{1}{\tau}m(S(\tau))=\pi m_0(X_0^*)

with {S(\tau)} denoting the slice {S(\tau)=\{M\in\mathcal{C}:\rho\geq\textrm{sys}(M)\geq\rho\exp(-\tau)\}}.

Next time, we will use this information and the Siegel-Veech formula to complete our solution of Eskin-Kontsevich-Zorich regularity conjecture.

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