Posted by: matheuscmss | July 22, 2011

Conférence internationale Géométrie Ergodique (Orsay 2011) II

For my second post related to the “Conférence Internationale Géométrie Ergodique” (held at the Math. Department of Orsay – Univ. Paris 11 from 23th to 27th May), I’ll transcript below my notes for the lecture of Manfred Einsiedler. The title of his talk was:

1. Escape of mass and Entropy

We begin by revising the following prototypical statement of “escape of mass” versus “entropy”:

Theorem 1 (M. E., E. Lindenstrauss, P. Michel, A. Venkatesh) Let ${\Gamma}$ be a subgroup of ${SL_2(\mathbb{R})}$ such that ${X=SL_2(\mathbb{R})/\Gamma}$ is non-compact and has finite volume. Let ${\mu_n}$ be a sequence of geodesic flow ${T_t:=diag(e^t,e^{-t})}$ invariant probability measures converging weakly-* to ${\mu_{\infty}}$. Then, the total mass of ${\mu_{\infty}}$ satisfies the inequality

$\displaystyle \mu_{\infty}(X)\geq 2\left(\limsup h_{\mu_n}(T)-\frac{1}{2}\right)$

where ${T=T_1}$. In other words, a sequence of geodesic flow invariant probabilities whose entropy are “high” can’t loose all its mass in the weak-* limit.

The reader can found more details on this theorem here. This result is inspired by the following classical equidistribution result (derived from the works of Yu. Linnik, B. Skubenko and W. Duke):

Theorem 2 (Linnik, Skubenko, Duke) Let ${X=SL_2(\mathbb{R})/SL_2(\mathbb{Z})}$ and ${K}$ be a real quadratic field. Let ${\mu_K}$ be the normalized Lebesgue measure on the finite collection of periodic orbits of the geodesic flow ${diag(e^t,e^{-t})}$ arising from the order determined by (the ideal classes of) the ring of intergers${\mathcal{O}_K\subset K}$ of the quadratic field ${K}$. Then,

$\displaystyle \mu_K\rightarrow m_X$

as “${K\rightarrow\infty}$”, i.e., ${\mu_K\rightarrow m_X}$ as the discriminant of ${K}$ goes to infinity. Here ${m_X}$ is the Haar measure on ${X}$.

Remark 1 See this paper of M. Einsiedler, E. Lindenstrauss, P. Michel and A. Venkatesh (Corollary 4.4) for more details on the fact that all periodic orbits of the geodesic flow on ${SL_2(\mathbb{R})/SL_2(\mathbb{Z})}$ can be obtained from ideal classes of the orders of real quadratic fields. Intuitevely, the idea is the following: it is well-known that the geodesic flow ${a_t=\textrm{diag}(e^t,e^{-t})}$ on the modular curve ${SL_2(\mathbb{R})/SL_2(\mathbb{Z})}$ is intimately related to the continued fraction algorithm; more precisely, given a geodesic in the hyperbolic plane ${\mathbb{H}}$, the way it crosses an appropriately fixed fundamental domain of the action of ${SL_2(\mathbb{Z})}$ on ${\mathbb{H}}$ is related to the continued fraction expansion of the “visual point” in the boundary of ${\mathbb{H}}$ where the geodesic “terminates”. In this way, it is not hard to convince ourselves that a geodesic can be periodic if and only if its successive passages through fundamental domains is periodic, that is, the continued fraction of the visual point is periodic. Now, it is also well-known that any real number with periodic continued fraction must be an quadratic irrational (i.e., they must belong to the beginning of the Markov and Lagrange spectrum), so we “see” that periodic geodesic of the modular curve has something to do with quadratic irrationals.

Recently, M. Einsiedler, S. Kadyrov and A. Pohl were able to prove the following “escape of mass vs entropy” results:

Theorem 3 (E., Kadyrov, Pohl) Let ${G}$ be a simple Lie group with real rank1, and let ${\Gamma}$ be a non-uniform lattice (i.e., ${G/\Gamma}$ is non-compact but with finite volume). Denote by ${T_t}$ the (1-parameter) Cartan subgroup of ${G}$, and let ${\mu_n}$ be a sequence of ${T_t}$ invariant probabilities converging (weakly-*) to ${\mu_{\infty}}$. Then,

$\displaystyle \frac{1}{2}(1-\mu_{\infty}(X))h_{\max}(T) + \mu_{\infty}(X) h_{\frac{\mu_{\infty}}{\mu_{\infty}(X)}}(T)\geq\limsup\limits_{n\rightarrow\infty}h_{\mu_n}(T),$

where ${T=T_s}$ for any ${s\in\mathbb{R}-\{0\}}$ (e.g., ${T=T_1}$).

Theorem 4 (E., Kadyrov) In the case ${G=SL_3(\mathbb{R})}$ and ${\Gamma=SL_3(\mathbb{Z})}$, let

$\displaystyle a_t:=\left(\begin{array}{ccc}e^{-t} & 0 & 0 \\ 0 & e^{t/2} & 0 \\ 0 & 0 & e^{t/2}\end{array}\right)$

and let ${\mu_n}$ be a sequence of ${a_t}$ invariant probability measures converging weakly-* to ${\mu_{\infty}}$. Then,

$\displaystyle \mu_{\infty}(X)\geq\limsup\limits_{n\rightarrow\infty}h_{\mu_n}(a_1)-2$

Remark 2 Concerning the “quality” of the bounds on ${\mu_{\infty}(X)}$ in the previous two theorems, let’s try to see what each bound gives. Firstly, we observe that the bound in Theorem 3gives that

$\displaystyle \begin{array}{rcl} &&(1-\mu_{\infty}(X))\frac{h_{\max}(T)}{2} + \mu_{\infty}(X) h_{\max}(T)\\ &\geq& (1-\mu_{\infty}(X))\frac{h_{\max}(T)}{2} + \mu_{\infty}(X) h_{\frac{\mu_{\infty}}{\mu_{\infty}(X)}}(T)\\&\geq& \limsup\limits_{n\rightarrow\infty}h_{\mu_n}(T), \end{array}$

that is,

$\displaystyle \mu_{\infty}(X)\geq\frac{2}{h_{\max}(T)}\limsup\limits_{n\rightarrow\infty}h_{\mu_n}(T)-1$

Secondly, even though the assumptions of the previous two theorems are not quite the same (e.g., the former deals with Lie groups with real rank 1 while the latter treats ${SL_3(\mathbb{R})}$, a Lie group of real rank 2), let’s pretend that we can use this in the situation of Theorem 4. In this case, since ${T=a_1}$ has maximal entropy ${h_{\max}(a_1)=3}$, the previous bound (“obtained” from Theorem 3) “ensures”

$\displaystyle \mu_{\infty}(X)\geq\frac{2}{3}\limsup\limits_{n\rightarrow\infty}h_{\mu_n}(T)-1.$

On the other hand, the bound provided by Theorem 4gives

$\displaystyle \mu_{\infty}(X)\geq \limsup\limits_{n\rightarrow\infty}h_{\mu_n}(T) - 2$

Since ${\frac{2}{3}x-1\geq x-2}$ whenever ${x\leq 3}$, and ${\limsup\limits_{n\rightarrow\infty}h_{\mu_n}(a_1)\leq h_{\max}(a_1)=3}$, we see that, in principle, the escape of mass estimate in Theorem 3 is “better” than the escape of mass estimate in Theorem 4 (as it was remarked by M. Einsiedler during his talk).

Among the applications of these results, M. Einsiedler mentioned sharp upper bounds for the Hausdorff dimension of singular vectors. We recall the definition of a singular vector below:

Definition 5 ${v\in\mathbb{R}^2}$ is singular if for all ${\varepsilon>0}$ and for all ${Q}$ large enough (i.e., ${Q>Q_0=Q_0(\varepsilon)}$), there are ${q<\varepsilon Q}$ and ${\underline{p}\in\mathbb{Z}^2}$ such that

$\displaystyle \|qv-\underline{p}\|<\varepsilon Q^{-3/2}$

The following theorem of Y. Cheung gives the exact Hausdorff dimension of the set of singular vectors of the plane.

Theorem 6 (Cheung) The Hausdorff dimension of the set of singular vectors in ${\mathbb{R}^2}$ is ${4/3}$.

As M. Einsiedler pointed out, Theorem 4 can be used to give the (sharp) upper bound ${\leq 4/3}$ on the Hausdorff dimension of singular vectors independently of the proof of Y. Cheung. Indeed, the link between singular vectors and dynamics on ${SL_3({\mathbb R})/SL_3({\mathbb Z})}$ is the following. It possible to show that ${v=(v_1,v_2)}$ is singular if and only if

$\displaystyle a_t x_v:=a_t \left(\begin{array}{ccc} 1 & 0 & 0 \\ v_1 & 1 & 0 \\ v_2 & 0 & 1\end{array}\right){\mathbb Z}^3\stackrel{t\rightarrow\infty}{\longrightarrow}\infty \quad\textrm{in} \quad X_3:=SL_3({\mathbb R})/SL_3({\mathbb Z}),$

i.e., the ${a_t}$-orbit of ${x_v}$ is diverging in ${X_3}$. Furthermore, one can check that the previous result on the Hausdorff dimension of singular vectors is equivalent to show that the Hausdorff dimension of the set of divergent points ${x}$ in ${X_3}$ (in the sense above) has Hausdorff dimension ${6+4/3}$. In particular, if one is only looking for upper bounds on the set of singular vectors, it suffices to estimate the set of diverging points in ${X_3}$. Since the set of diverging points in ${X_3}$ is a subset of the set of points ${x}$ diverging on average, i.e.,

$\displaystyle \frac{1}{T}\int_0^T f(a_t x) \rightarrow 0$

for every smooth compactly supported ${f}$ (or in other words, the mass of the orbit is escaping to infinity), we can limit ourselves to estimate the Hausdorff dimension of the set of points diverging on average. Now, the Hausdorff dimension of this set can be estimate from the “entropy” result of Theorem 4 in view of the works of Ledrappier and coauthors. For instance, in the particular case of

$\displaystyle a_t=\left(\begin{array}{ccc} e^{-t} & 0 & 0 \\ 0 & e^{t/2} & 0 \\ 0 & 0 & e^{t/2}\end{array}\right)$

there are only two directions of expansion, i.e., the unstable manifold is

$\displaystyle \left(\begin{array}{ccc} 1 & 0 & 0 \\ \ast & 1 & 0 \\ \ast & 0 & 1\end{array}\right).$

Moreover, the particular choice of ${a_t}$ using ${e^{t/2}}$ along the main diagonal (instead of ${e^{\alpha t}}$, ${e^{\beta t}}$ with ${\alpha+\beta=1}$) implies that ${a_t}$ expands the two directions at the same speed (i.e., ${a_t}$ is conformalalong the unstable direction), so that it behaves “like a one-dimensional map” along the unstable and hence the application of Ledrappier et al. work (to deduce Hausdorff dimension bounds from entropy) is easier.

Next, M. Einsiedler tried to make this relationship between Hausdorff dimension and entropy more clear by giving a sketch of proof of the following result of Y. Cheung:

Theorem 7 (Cheung) The set of elements of ${X_2:=SL_2({\mathbb R})/SL_2({\mathbb Z})}$ that are diverging on average has Hausdorff dimension ${2+1/2}$.

As we’re going to see below, M. Einsiedler’s choice of ${X_2}$ instead of ${X_3}$ was motivated by the fact that the unstable dimension is really 1-dimensional in the case of ${X_2}$, while it is conformal but not exactly 1-dimensional in the case of ${X_3}$ (so that, even though there are no philosophical distinction between these case, the life is technically easier in the former case).

In fact, Manfred’s sketch of proof of the previous theorem of Cheung aims to “explain” why the factor ${1/2}$ appears. To do so, recall that ${X_2:=SL_2({\mathbb R})/SL_2({\mathbb Z})}$ is the space of unimodular lattices on ${\mathbb{R}^2}$ and take ${X\in X_2}$ a lattice “going up” under geodesic flow on ${X_2}$ (viewed as the unit cotangent bundle of the quotient of the hyperbolic plane ${\mathbb{H}}$ by ${SL_2({\mathbb Z})}$), and suppose

$\displaystyle Y=\left(\begin{array}{cc} 1 & 0 \\ s & 1 \end{array}\right)X$

is also “going up” and coming back at a compact part at the same time as ${X}$.

By taking ${(a,b)\in X}$ a “short” vector (such a vector must exist by Mahler’s compactness criterion [otherwise we couldn’t “go up”]), we know that “going up” and coming back to a compact part at time ${t=n}$ for ${X}$ means that

$\displaystyle \left(\begin{array}{cc} e^{-t/2} & 0 \\ 0 & e^{t/2} \end{array}\right)\left(\begin{array}{c} a \\ b \end{array}\right)$

is short until time ${t=n}$, i.e., ${|a|\gg 1}$ and ${|b|\sim e^{-n/2}}$. Now, the analogous statement for ${Y}$ gives ${|e^{n/2}(b+sa)|\leq 1}$, i.e., ${|s|\ll e^{-n/2}}$.

On the other hand, the Bowen (dynamical) balls of time ${n}$ (i.e., the set of points whose trajectories up to time ${n}$ are “close”) have size ${\sim e^{-n}}$ along the unstable direction

$\displaystyle \left(\begin{array}{cc} 1 & 0 \\ \ast & 1 \end{array}\right)$

In this way, the factor ${1/2}$ in Cheung’s theorem exists to take into account the logarithmic difference between ${e^{-n/2}}$ and ${e^{-n}}$. Of course, this heuristic can be formalized by taking into account that counting Bowen balls is directly related to entropy, while covering a given set (in this case the set of points diverging on average) by “balls” is related to Hausdorff dimension, and the precise relation between entropy and dimension is the core of the so-called Ledrappier-Young formula.

At this point, M. Einsiedler started running out of time, so he gave the following crude outline of proof of his results with S. Kadyrov and A. Pohl. The basic idea is to use a nice idea appearing in the works of Eskin and Margulis, and Eskin, Margulis and Mozes, namely, one can carefully introduce a proper (“height”) function ${\Delta:X_n\rightarrow\mathbb{R}_+}$ (where ${X_n:=SL_n({\mathbb R})/SL_n({\mathbb Z})}$) such that

$\displaystyle \int_Q \Delta(a_t u(s)x)ds\leq c\Delta(x)+D$

where ${Q}$ is a piece of unstable manifold, ${c<1}$ and ${D}$ is some constant (which is irrelevant when we’re “high up” in the cusp).

Remark 3 In the original context of Eskin and Margulis (i.e., the study of recurrence properties), the estimate they derived was of the form

$\displaystyle \int \Delta(...x)ds\leq c\Delta(x)+D$

In other words, when compared to this estimate of Eskin and Margulis, the technical novelty introduced by Einsiedler and his coauthors is the integration along pieces of unstable manifolds only.

Then, one shows that this estimate can be converted into the following information about Bowen balls: the number of Bowen balls inside ${Q}$ which have height ${\geq\Delta(x)}$ after application of ${a_t}$ is bounded by ${c^n}$ times the total number of Bowen balls inside ${Q}$. Finally, once one has a nice control of Bowen balls “coming back” to a compact part, the actual escape of mass can be estimated in terms of entropy (that is, asymptotics of Bowen balls).

At this point, M. Einsiedler ended his talk. Closing this post, let me just make the following “historical” remark:

Remark 4 In Teichmüller dynamics, the same technique as above (i.e., trying to control an appropriate proper function ${\Delta}$ to get good dynamical properties) was used by Eskin and Masur to get quadratic asymptotics to number of saddle connections in translation surfaces, and by Avila and Forni to perform parameter exclusions and derive the weak-mixing property for typical interval exchange maps and translation flows. Finally (and more recently), this idea was retaken by Benoist and Quint in their study of stationary measure for homogenous dynamics.