Posted by: matheuscmss | March 12, 2013

Eskin-Kontsevich-Zorich regularity conjecture IV: a perfect cancellation result and end of proof of EKZ conjecture

Today we will complete the description of the solution of Eskin-Kontsevich-Zorich regularity conjecture, that is, we will prove that, for any {SL(2,\mathbb{R})}-invariant probability measure {m} on a connected component of a stratum of the moduli space of translation surfaces of genus {g\geq 2}, the {m}-measure of the set {\mathcal{C}_2(\rho)} of {M\in\mathcal{C}} with two non-parallel saddle-connections of lengths {\leq\rho} is

\displaystyle m(\mathcal{C}_2(\rho))=o(\rho^2)

For this sake, let us recall that, in the previous post of this series, we considered an arbitrarily fixed level {X=\{M\in\mathcal{C}: \textrm{sys}(M)=\rho\}} of the systole function and we introduced a subset {X_0^*\subset X} consisting of {M\in X} such that all its non-vertical saddle-connections have length {>\rho}. Then, we defined the set

\displaystyle Y^*=\{g_tR_{\theta}M_0: M_0\in X_0^*, |\theta|<\pi/4, 0<t<\log\cot|\theta|\}

and we studied the {m}-measure of the subsets

\displaystyle Y^*(T):=Y^*\cap\{M\in\mathcal{C}:\textrm{sys}(M)<\rho\exp(-T)\}

of translation surfaces with systole {<\rho\exp(-T)} that are “accessible” by {g_t} and {R_{\theta}} movements from {X_0^*}. In this setting, the results of the previous post of this series can be summarized as follows:

  • {m|_{Y^*}=\cos2\theta dt\,d\theta\,m_0}, where {m_0} is a finite measure on {X_0^*};
  • {m(Y^*(T))=(1/2)\pi\,m_0(X_0^*)\exp(-2T)} for all {T>0};
  • {m_0} is a density measure in the sense that

    \displaystyle \pi\,m_0(X_0^*)=\lim\limits_{\tau\rightarrow0}\frac{1}{\tau}m(\{M\in\mathcal{C}:\rho\geq\textrm{sys}(M)\geq\rho\exp(-\tau)\})

From this point, we will divide this final post into two sections. In the first one, we will formalize the idea that {Y^*(T)} occupies most of {\{M\in\mathcal{C}:\textrm{sys}(M)\leq\rho\exp(-T)\}} for adequate choices of {\rho}, so that the proof of Eskin-Kontsevich-Zorich regularity conjecture will be reduced to the computation of the {m}-measure of {\mathcal{C}_2(\rho\exp(-T))\cap Y^*(T)}. Then, in the final section, we will show that the set of {M_0\in X_0^*} such that {g_t R_{\theta} M_0\in\mathcal{C}_2(\rho\exp(-T))\cap Y^*(T)} has small {m}-measure for {T>0} large because these {M_0}‘s have a pair of non-parallel saddle-connections with a small angle (and {m|_{Y^*}} is {\cos2\theta dt\, d\theta\,m_0}).

1. A “perfect cancellation” result

Let {F(\rho):=m(\{M\in\mathcal{C}:\textrm{sys}(M)\leq\rho\})}. From the fact that {m|_{Y^*}=\cos2\theta dt d\theta m_0} imply that the level sets of the systole function have zero {m}-measure, and, thus, {F} is a continuous non-increasing function.

Furthermore, the fact that {m_0} is a density measure implies that {F} has a left derivative

\displaystyle F'(\rho)=\lim\limits_{\tau\rightarrow0}\frac{F(\rho)-F(\rho\exp(-\tau))}{\rho-\rho\exp(-\tau)}

equal to {\pi m_0(X_0^*)/\rho}.

By combining this with the facts that {Y^*(T)\subset \{M\in\mathcal{C}: \textrm{sys}(M)\leq\rho\exp(-T)\}} and {m(Y^*(T))=(1/2)\pi m_0(X_0^*) \exp(-2T)}, we get that

\displaystyle \frac{1}{2}\rho F'(\rho)\exp(-2T)=\frac{\pi}{2} m_0(X_0^*)\exp(-2T) = m(Y^*(T))\leq F(\rho\exp(-T)) \ \ \ \ \ (1)

Moreover, the Siegel-Veech formula says that {F(s)=O(s^2)} for all {s>0}, so that the previous estimate imply that

\displaystyle F'(\rho)=O(\rho)

In particular, the left derivative {F'} is bounded.

From this, it follows that {F} is absolutely continuous: indeed, if the left derivative {F'} of a continuous function {F} is bounded by {C} on an interval {[\rho_0,\rho_1]}, then {|F(\rho_1)-F(\rho_0)|\leq C(\rho_1-\rho_0)} because, for any {C'>C}, one can check that the supremum of the {\rho\in[\rho_0,\rho_1]} such that {|F(\rho)-F(\rho_0)|\leq C(\rho-\rho_0)} is {\rho_1} (in view of the assumption that {F'(\rho)\leq C} for all {\rho\in[\rho_0,\rho_1]}).

At this point, we are ready to “improve” the Siegel-Veech formula by showing that there exists a constant {C(m)>0} such that

\displaystyle \lim\limits_{\rho\rightarrow0} F(\rho)/\rho^2=C(m)

Indeed, let

\displaystyle c(m):=\sup\limits_{\rho \textrm{ s.t. }F(\rho)<1} F'(\rho)/\rho

Since {F} is absolutely continuous, it is the integral of its left-derivative. It follows that

\displaystyle F(\rho)=\int_0^{\rho} F'(s)\,ds = \int_0^{\rho}\frac{F'(s)}{s}s\,ds\leq \frac{1}{2}c(m) \rho^2

for all {\rho>0}, and, hence,

\displaystyle \limsup\limits_{\rho\rightarrow0}F(\rho)/\rho^2\leq (1/2)c(m).

On the other hand, from (1), we know that

\displaystyle \frac{1}{2}\frac{F'(\rho)}{\rho}\rho^2\exp(-2T)\leq F(\rho\exp(-T))

for all {T>0}. Therefore, by taking {s=\rho\exp(-T)}, we obtain that

\displaystyle (1/2)c(m)\leq \liminf\limits_{s\rightarrow0}F(s)/s^2.

In summary, we showed that

\displaystyle \lim\limits_{\rho\rightarrow0}\frac{F(\rho)}{\rho^2}=(1/2)c(m)

Once we know that {F(\rho)=((1/2)c(m)+o(1))\rho^2}, we can show the following “perfect cancellation” equality:

\displaystyle \limsup\limits_{\rho\rightarrow0}\frac{F'(\rho)}{\rho}=c(m)

In fact, it is obvious that {\limsup\limits_{\rho\rightarrow0}\frac{F'(\rho)}{\rho}\leq c(m)}, and, for the converse inequality, we can use that {F} is the integral of its left derivative:

\displaystyle (1/2)c(m)=\lim\limits_{\rho\rightarrow0}F(\rho)/\rho^2= \lim\limits_{\rho\rightarrow0}\left(\int_0^{\rho}(F'(s)/s)s\,ds\right)/\rho^2

\displaystyle \leq (1/2)\limsup\limits_{s\rightarrow 0}F'(s)/s

We call the equality

\displaystyle \limsup\limits_{\rho\rightarrow0}\frac{F'(\rho)}{\rho}=c(m)

a “perfect cancellation” by the following reason. Recall that {F'(\rho)/\rho=\pi m_0(X_0^*)/\rho^2} (where {m_0=m_0(\rho)} and {X_0^*=X_0^*(\rho)}). So, by taking {\rho_n\rightarrow0} a sequence such that {F'(\rho_n)/\rho_n=c(m)+o(1)} as {n\rightarrow\infty}, and by considering the sets {Y^*(T)=Y^*(\rho_n,T)}, we conclude that

\displaystyle m(Y^*(\rho_n,T))=\frac{1}{2}\frac{\pi m_0(X_0(\rho_n))}{\rho_n} \exp(-2T)=\frac{1}{2}\frac{F'(\rho_n)}{\rho_n}\rho_n^2\exp(-2T)

\displaystyle = \frac{1}{2}(c(m)+o(1))(\rho_n\exp(-T))^2=F(\rho_n\exp(-T)).

Since {m(\{M\in\mathcal{C}: \textrm{sys}(M)\leq\rho_n\exp(-T)\}):=F(\rho_n\exp(-T))}, we conclude that {Y^*(\rho_n, T)} occupies most of {\{M\in\mathcal{C}: \textrm{sys}(M)\leq\rho_n\exp(-T)\}} in the sense that

\displaystyle m(\{M\in\mathcal{C}: \textrm{sys}(M)\leq\rho_n\exp(-T)\}-Y^*(\rho_n, T))=o(1)(\rho_n\exp(-T))^2

In particular, our task of showing that

\displaystyle m(\mathcal{C}_2(\rho))=o(\rho^2)

is reduced to show that, for each {\rho_n}, {n\in\mathbb{N}} and all {T>0} sufficiently large (depending on {\rho_n}),

\displaystyle m(\mathcal{C}_2(\rho_n\exp(-T))\cap Y^*(\rho_n, T))=o((\rho_n\exp(-T))^2)

The proof of this last assertion will occupy the next (last) section of this post.

2. End of proof of EKZ regularity conjecture

Let us fix once and for all one of the {\rho_n}‘s, {n\in\mathbb{N}}, as above, and let us consider the set {C_2(\rho_n\exp(-T))\cap Y^*(\rho_n, T))}. By definition, they consist of {M\in\mathcal{C}} of the form {M=g_t R_{\theta}M_0} with {M_0\in X_0^*(\rho_n)} and {t\in J(T,\theta)} possessing two non-parallel saddle-connections of lengths {\leq\rho_n\exp(-T)}.

Without loss of generality, we can assume that the lengths of these two saddle-connections are comparable within a large multiplicative factor {A>1}. Indeed, if {A} is very large and the ratio of these saddle-connections is larger than {A}, then the systole of {M} is actually {\leq (1/A)\rho_n\exp(-T)} and, by the Siegel-Veech formula, the {m}-measure of the set

\displaystyle \{M\in \mathcal{C}:\textrm{sys}(M)\leq (1/A)\rho_n\exp(-T)\}

is {O((1/A^2)\rho_n^2\exp(-2T))=o(\rho_n^2\exp(-2T))} for {A} large enough.

Next, we consider the set of {M_0\in X_0^*=X_0^*(\rho_n)} giving birth to {M=g_t R_{\theta} M_0\in \mathcal{C}_2(\rho_n\exp(-T))} with a pair of saddle-connections of lengths comparable within a large multiplicative factor {A>1} from the systole.

We affirm that, for {T>0} large enough, the angle {\theta} is small, that is, for each {\omega_0>0} and {\alpha_0}, there are {T_0=T_0(\alpha_0)} and {K=K(\omega_0)>0} such that if {M=g_t R_{\theta} M_0\in \mathcal{C}_2(\rho_n\exp(-T))} with {M_0\in X_0^*} and {T\geq T_0}, then:

  • either {|\sin2\theta|\leq\exp(-2T)\sin\omega_0}
  • or {M_0} has a pair of non-parallel saddle-connections of lengths {\leq A K\textrm{sys}(M_0)} making an angle {<\alpha_0}.

Assuming momentarily this claim, let us complete the proof of EKZ regularity conjecture. In this direction, let us consider the first item above and let us show that the {m}-measure of this event is {o(\rho_n^2\exp(-2T))} if {\omega_0} small enough. Recall that {m|_{Y^*}=\cos2\theta dt \,d\theta\,m_0}. Therefore, given {\omega_0>0}, for all {T>0}, the {m}-measure of the subset of translation surfaces in {Y^*(T)} with {|\sin2\theta|\leq\exp(-2T)\sin\omega_0} is

\displaystyle m_0(X_0^*)\int_{-\omega_0}^{\omega_0}|J(T,\theta)|\cos2\theta\,d\theta

\displaystyle =(1/4)\pi\,m_0(X_0^*)\exp(-2T)\int_{-\omega_0}^{\omega_0}\log\frac{1+\cos\omega}{1-\cos\omega}\cos\omega d\omega.

Since this quantity is {o(\rho_n^2\exp(-2T))} for {\omega_0>0} small enough, it suffices to estimate the {m}-measure of the set of {M=g_t R_{\theta} M_0} with {M_0\in B(\alpha_0)} where {B(\alpha_0)} denotes the event described in the second item of our claim. Here, we observe that, by discreteness of the set of holonomy vectors of saddle-connections, {m_0(B(\alpha_0))} is {o(1)} for {\alpha_0} small enough. Therefore, the {m}-measure of the set of {M=g_t R_{\theta} M_0} with {M_0\in B(\alpha_0)} is

\displaystyle m_0(B(\alpha_0))\int_{-\pi/4}^{\pi/4}|J(T,\theta)|\cos2\theta d\theta = \frac{1}{2}\pi m_0(B(\alpha_0))\exp(-2T)

\displaystyle =o(\rho_n^2\exp(-2T)),

so that the proof of EKZ regularity conjecture is complete.

Now, let us prove our affirmation. Since {M_0\in X_0^*}, {M} has a saddle-connection in the direction of {g_t R_{\theta} e_2}. Now, let us try to figure out what non-vertical vectors {v\in\mathbb{R}^2} can represent (the holonomy vector of) a saddle-connection on {M_0} such that the saddle-connection of {M} represented by {g_t R_{\theta} v} has length comparable to {g_t R_{\theta} e_2} within the multiplicative factor {A>1}.

Suppose that the first item of our claim is violated, i.e., {\sin\omega_0\exp(-2T)<|\sin2\theta|}. In this situation, as we saw in Subsection “On the action of the diagonal subgroup {g_t=\textrm{diag}(e^t, e^{-t})}” of the second post of this series, given {\omega_0}, there exists a constant {K=K(\omega)>0} such that if {\sin\omega_0\exp(-2T)<|\sin2\theta|<\exp(-2T)}, then {\|g_t R_{\theta} e_2\|\leq K\exp(-t)} for all {t\in J(T,\theta)}. It follows that, if {\|v\|>AK\rho_n}, then

\displaystyle \|g_t R_{\theta} v\|> AK\rho_n\exp(-t)\geq A\|g_t R_{\theta}(\rho_n e_2)\|.

In other words, if we start with a saddle-connection of {M_0} whose holonomy vector {v\in\mathbb{R}^2} satisfies {\|v\|>AK\rho_n}, then {g_t R_{\theta} v} is not comparable to {g_t R_\theta (\rho_n e_2)} within the multiplicative factor {A>1}. Equivalently, if {v} is the holonomy vector of a saddle-connections of {M_0} leading to saddle-connections of {M=g_t R_{\theta}M_0} whose lengths are between {\textrm{sys}(M)} and {A\textrm{sys}(M)}, then {\|v\|\leq AK\rho_n=AK\textrm{sys}(M_0)}.

At this point, our task consists into checking that if {v} with {\|v\|\leq AK\textrm{sys}(M_0)} is the holonomy vector of a saddle-connections of {M_0} leading to saddle-connections of {M=g_t R_{\theta}M_0} whose lengths are between {\textrm{sys}(M)} and {A\textrm{sys}(M)}, then the angle between {v} and {e_2} is {<\alpha_0} if {T} is large enough and {t\in J(T,\theta)}. However, this last property is not hard to get: if {\rho_n\leq \|v\|\leq A K\rho_n} and the angle between {v} and {e_2} is {\geq\alpha_0}, then

\displaystyle g_t R_{\theta} v= \|v\|g_t R_{\theta+\theta'}e_2

for some {\theta'\geq\alpha_0}. Since {g_t} expands horizontal vectors, one can check that

\displaystyle \|g_t R_{\theta+\theta'} e_2\|>A\|g_t R_{\theta} e_2\|

for {\theta'\geq\alpha_0>0} and {t} large enough depending on {\alpha_0}, e.g., {t\in J(T,\theta)} and {T\geq T_0=T_0(\alpha_0)}. In other terms, if {t\in J(T,\theta)} with {T\geq T_0(\alpha_0)}, then a saddle-connection of {M=g_t R_{\theta} M_0} ({M_0\in X_0^*}) of length {\leq A \textrm{sys}(M)} non-parallel to a length-minimizing saddle-connection of {M} must come from a saddle-connection of {M_0} of length {\leq AK\textrm{sys}(M_0)} making an angle {<\alpha_0} with the vertical direction.

In summary, we showed that if the first item of our claim is violated, then the second item holds. This proves the claim and, a fortiori, the proof of EKZ regularity conjecture.


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