Recall that the main goal of this series of posts is the proof of the following result:

Theorem 1 Let {G} be a semisimple algebraic Lie group of real rank {\textrm{rank}_{\mathbb{R}}(G)\geq 2}. Denote by {U\subset G} a horospherical subgroup of {G}. If {\Gamma\subset G} is a discrete Zariski-dense and irreducible subgroup such that {\Gamma\cap U} is cocompact, then {\Gamma} is commensurable to an arithmetic lattice {G_{\mathbb{Z}}}.

Last time, we discussed the first half of the proof of this theorem in the particular case of {G=SL(2p,\mathbb{R})}, {p\geq 2}, and {U=\left\{\left( \begin{array}{cc} I & B \\ 0 & I \end{array} \right): B\in M(p,\mathbb{R})\right\}}. Actually, we saw that this specific form of {U\subset G} is not very important: all results from the previous post hold whenever

  • {U} is reflexive: in the context of the example above, this is the fact that {U} is conjugate to the opposite horospherical subgroup {U^-=\left\{\left( \begin{array}{cc} I & 0 \\ C & I \end{array} \right): C\in M(p,\mathbb{R})\right\}};
  • {U} is commutative.

Indeed, we observed that {U} reflexive allows to also assume that {\Gamma\cap U^-} is cocompact in {U^-}. Then, this property and the commutativity of {U} were exploited to establish the closedness of the {\textrm{Ad}_L}-orbit of {(\Lambda, \Lambda^-)} in {X_{\mathfrak{u}}\times X_{\mathfrak{u}^-}}, where {\Lambda:=\log(\Gamma\cap U)\in X_{\mathfrak{u}} = \{\textrm{lattices in }\mathfrak{u}\}}, {\Lambda^-:=\log(\Gamma\cap U^-)\in X_{\mathfrak{u}^-} = \{\textrm{lattices in }\mathfrak{u}^-\}}, and

\displaystyle L=P\cap P^- = \left\{\left(\begin{array}{cc} A & 0 \\ 0 & D \end{array}\right):\textrm{det}(A)\cdot\textrm{det}(D)=1\right\}

is the common Levi subgroup of the parabolic subgroups {P=N_G(U)} and {P^-=N_G(U^-)} normalizing {U} and {U^-}.

Today, we will discuss the second half of the proof of Theorem 1 in the particular case of {G=SL(2p,\mathbb{R})}, {p\geq 2}, and {U=\left\{\left( \begin{array}{cc} I & B \\ 0 & I \end{array} \right): B\in M(p,\mathbb{R})\right\}}: in other terms, our goal below is to obtain the arithmeticity of {\Gamma} from the closedness of {\textrm{Ad}_L(\Lambda, \Lambda^-)} in the homogenous space {X_{\mathfrak{u}}\times X_{\mathfrak{u}^-} := G_0/\Gamma_0}. This step is due to Hee Oh (see Proposition 3.4.4 of her paper).

1. From closedness to infinite stabilizer

Let {S=[L,L] = \left\{ \left(\begin{array}{cc} A & 0 \\ 0 & D \end{array}\right)\in G: \textrm{det}(A) = \textrm{det}(D) = 1 \right\} \simeq SL(p,\mathbb{R})\times SL(p,\mathbb{R})} and {A=\left\{ \left(\begin{array}{cc} \lambda I & 0 \\ 0 & \lambda^{-1} I \end{array}\right)\in G: \lambda\in\mathbb{R}^*\right\}}, so that {L=AS}. In particular, the closedness of {\textrm{Ad}_L(\Lambda, \Lambda^-)} implies that

{\textrm{Ad}_S(\Lambda, \Lambda^-)} is closed in {X_{\mathfrak{u}}\times X_{\mathfrak{u}^-} = G_0/\Gamma_0}.

The next proposition asserts that the stabilizer of this orbit is large whenever {S} is not compact:

Proposition 2 The stabilizer {\textrm{Stab}_S(\Lambda,\Lambda^-)=\{s\in S: \textrm{Ad}_s(\Lambda,\Lambda^-) = (\Lambda,\Lambda^-)\}} is a lattice in {S}.

This proposition is a direct consequence of the closedness of the {\textrm{Ad}_S(\Lambda, \Lambda^-)} in {G_0/\Gamma_0} and the following general fact:

Proposition 3 Let {G_0} be a Lie group, {\Gamma_0} a lattice in {G_0}, and {x_0\in X_0=G_0/\Gamma_0}. Suppose that {S_0\subset G_0} is a semisimple subgroup with finite center such that {S_0 x_0} is closed, then {S_0\cap \Gamma_0} is a lattice in {S_0}.

Proof: The first ingredient of the argument is Howe–Moore’s mixing theorem: it asserts that if {S_0} is a semisimple group with finite center and {(\mathcal{H}_0, \pi_0)} is an unitary representation of {S_0} with {\{v\in\mathcal{H}_0: \pi_0(S_0)v = v\}=\{0\}}, then

\displaystyle \lim\limits_{s\rightarrow\infty}\langle\pi_0(s) v_0, w_0\rangle = 0

for all {v_0, w_0\in\mathcal{H}_0}. (Here, {s\rightarrow\infty} means that the projection of {s} to any simple factor {S^{(i)}} of {S_0 = \prod S^{(i)}} diverges.)

The second ingredient of the argument is Dani–Margulis recurrence theorem: it says that if {\Gamma_0} is a lattice in a Lie group {G_0} and {u_t} is a one-parameter unipotent subgroup of {G_0}, then, given {x_0\in X_0=G_0/\Gamma_0} and {\varepsilon>0}, there exists a compact subset {K\subset X_0} such that

\displaystyle \frac{1}{T}\textrm{Leb}(\{t\in [0,T]: u_t x_0\in K\}) \geq 1-\varepsilon

for all {T>0}.

The basic idea to obtain the desired proposition is to apply these ingredients to {\mathcal{H}_0 = L^2(X_0,\lambda_0)}, where {\lambda_0} is a {S_0}-invariant measure on {S_0 x_0}, and {u_t} is a one-parameter unipotent subgroup in the product {S_0''} of non-compact simple factors of {S_0} . Here, we observe that {\lambda_0} is a bona fide Radon measure because we are assuming that {S_0 x_0} is closed, and, if we take {u_t} not contained in proper normal subgroups of {S_0''}, then {u_t\rightarrow\infty} as {t\rightarrow\infty} thanks to the absence of compact factors in {S_0''}. In this setting, our task is reduced to prove that {\lambda_0} is a finite measure.

In this direction, we apply Dani–Margulis recurrent theorem to get {A\subset X_0} with {0 < \lambda_0(A) < \infty} and a compact subset {K\subset X_0} such that

\displaystyle \frac{1}{T}\textrm{Leb}(\{t\in [0,T]: u_t x\in K\}) \geq \frac{1}{2}

for all {T>0} and {x\in A}. In this way, we obtain that the characteristic functions {1_A} and {1_K} of {A} and {K} are two elements of {\mathcal{H}_0} with {\langle \pi_0(u_t) 1_A, 1_K \rangle = \lambda_0(A\cap u_t^{-1}(K))}, and, hence, by Fubini’s theorem,

\displaystyle \begin{array}{rcl} \frac{1}{T}\int_0^T \langle \pi_0(u_t) 1_A, 1_K \rangle\,dt &=& \frac{1}{T}\int_0^T \lambda_0(A\cap u_t^{-1}(K)) \,dt \\ &=& \int_A \frac{1}{T}\textrm{Leb}(\{t\in [0,T]: u_t x\in K\}) \, d\lambda_0 \\ &\geq& \lambda_0(A)/2 > 0 \end{array}

for all {T>0}. It follows from Howe–Moore’s mixing theorem that

\displaystyle \{v\in\mathcal{H}_0: \pi_0(S_0'')v=v\}\neq \{0\},

that is, there exists a non-zero function {\varphi''\in\mathcal{H}_0} which is {S_0''}-invariant. By averaging {\varphi''} over the product {S_0'} of the compact factors of {S_0} if necessary, we obtain a non-zero function {\varphi\in\mathcal{H}_0} which is {S_0}-invariant. By ergodicity of {\lambda_0}, we have that {\varphi} is constant and, a fortiori, {\lambda_0} is a finite measure. \Box

2. From infinite stabilizer to {\Gamma\cap L} infinite

Our previous discussions (about {\textrm{Ad}_L}-actions) paved the way to understand {\Gamma\cap L}. Intuitively, it is important to get some information about {\Gamma\cap L} in our way towards showing the arithmeticity of {\Gamma} because we already know that {\Gamma\cap U} and {\Gamma\cap U^-} are lattices (i.e., {\Gamma} projects to lattices in “other directions”).

The intuition in the previous paragraph is confirmed by Margulis construction of {\mathbb{Q}}-forms:

Theorem 4 (Margulis) If {\Gamma\cap L} is infinite, then {\Gamma} is contained in some {\mathbb{Q}}-form {G_{\mathbb{Q}}} of {G}.

We will discuss the proof of this result in the next section. For now, we want to exploit the information {\Gamma\subset G_{\mathbb{Q}}} in order to derive the arithmeticity of {\Gamma}. For this sake, we invoke the following result:

Theorem 5 (Raghunathan-Venkataramana) Assume that {G} is semisimple of {\textrm{rank}_{\mathbb{R}}(G)\geq 2} defined over {\mathbb{Q}}, and {G} is {\mathbb{Q}}-simple. Suppose that {U} and {U^-} are opposite horospherical subgroups defined over {\mathbb{Q}}.If {\Gamma\subset G_{\mathbb{Z}}} is a subgroup such that {\Gamma\cap U_{\mathbb{Z}}}, resp. {\Gamma\cap U_{\mathbb{Z}}^-}, has finite index in {U_{\mathbb{Z}}}, resp. {U_{\mathbb{Z}}^-}, then {\Gamma} has finite index in {G_{\mathbb{Z}}}.

Remark 1 As it was kindly pointed out to me by David Fisher, Raghunathan-Venkataramana theorem is due to Margulis in the cases covered by Raghunathan (at least).

Remark 2 This result is false when {\textrm{rank}_{\mathbb{R}}(G)=1}, e.g., {G=SL(2,\mathbb{R})} (and {\Gamma=\left\langle\left(\begin{array}{cc}1 & 3 \\ 0 & 1 \end{array}\right), \left(\begin{array}{cc}1 & 0 \\ 3 & 1 \end{array}\right) \right\rangle}).

Roughly speaking, Raghunathan-Venkataramana theorem essentially establishes the desired Theorem 1 provided we know in advance that {\Gamma\subset G_{\mathbb{Q}}}.

In the sequel, we will treat Raghunathan-Venkataramana theorem as a blackbox and we will complete the proof of Theorem 1 in the case {U} reflexive and commutative, and {S} non-compact such as {G=SL(2p,\mathbb{R})} and {U = \left\{\left(\begin{array}{cc} I & B \\ 0 & I\end{array}\right)\in G: B\in M(p,\mathbb{R})\right\}}.

Remark 3 In some natural situations (e.g., subgroups generated by the matrices of the so-called Kontsevich–Zorich cocycle) we have that {\Gamma\subset G_{\mathbb{Z}}}. In particular, it is a pity that the lack of time made that Yves Benoist could not explain to me the proof of Raghunathan-Venkataramana theorem. Anyhow, I hope to come back to discuss this point in more details in the future.

Proof of Theorem 1:  We consider the subgroup {\Gamma':=\langle \Gamma\cap U, \Gamma\cap U^-\rangle} of {\Gamma}. It is discrete and Zariski dense in {G}. Therefore, the normalizer {\Gamma''=N_G(\Gamma')} is Zariski dense in {G}, and it is not hard to check that it is also discrete.

Observe that Proposition 2 says that {\Gamma''\cap S} is a lattice in {S} (because {\Gamma''=N_G(\Gamma')\supset \textrm{Stab}_S(\Lambda, \Lambda^-)}). Since {S} is non-compact, we have that {\Gamma'\cap L} is infinite. Thus, Margulis’ Theorem 4 implies that {\Gamma'\subset\Gamma''\subset G_{\mathbb{Q}}} for some {\mathbb{Q}}-form of {G}.

Note also that {U} and {U^-} are defined over {\mathbb{Q}}: in fact, if {H\subset GL(n,\mathbb{R})} is an algebraic subgroup such that {H\cap GL(n,\mathbb{Q})} is Zariski dense in {H}, then {H} is defined over {\mathbb{Q}}.

Hence, we can apply Raghunathan-Venkataramana Theorem 5 to get that {\Gamma'} is commensurable to {G_{\mathbb{Z}}}. Since {\Gamma'\subset\Gamma}, this proves the arithmeticity of {\Gamma}. \Box

Remark 4 As we noticed above, the arguments presented so far allows to prove Theorem 1 when {U} is reflexive and commutative, and {S} is non-compact.

3. From {\Gamma\cap L} infinite to arithmeticity

In this (final) section (of this post), we discuss some steps in the proof of Margulis  Theorem 4 stated above.

We write {\mathfrak{g} = \mathfrak{u}^-\oplus \underbrace{\mathfrak{s}\oplus\mathfrak{a}}_{=\mathfrak{l}} \oplus\mathfrak{u}}, i.e., we decompose the Lie algebra of {G} in terms of the Lie algebras of {U^-}, {U} and {L=AS}.

Our goal is to find a {\mathbb{Q}}-form of {G} containing {\Gamma}. For this sake, let us do some “reverse engineering”: assuming that we found {G_{\mathbb{Q}}\supset\Gamma}, what are the constraints satisfied by the {\mathbb{Q}}-structure {\mathfrak{g}_{\mathbb{Q}}} on its Lie algebra?

First, we note that we dispose of lattices {\Lambda\subset \mathfrak{u}} and {\Lambda^-\subset\mathfrak{u}^-}. Hence, we are “forced” to define {\mathfrak{u}_{\mathbb{Q}}} and {\mathfrak{u}^-_{\mathbb{Q}}} as the {\mathbb{Q}}-vector spaces spanned by {\Lambda} and {\Lambda^-}.

Next, we observe that the choice of {\mathfrak{u}_{\mathbb{Q}}} above imposes a natural {\mathbb{Q}}-structure {\mathfrak{s}_{\mathbb{Q}}} on {\mathfrak{s}} via the adjoint map {\textrm{Ad}:S\rightarrow SL(\mathfrak{u})}. In fact, {S} is defined over {\mathbb{Q}} because we know that {\Gamma\cap S} is a lattice (and, hence, Zariski-dense) in {S} and {\textrm{Ad}(S\cap\Gamma)\subset SL(\mathfrak{u}_{\mathbb{Q}})} (by definition).

Finally, since {U}, {U^-} and {S} are already defined over {\mathbb{Q}} and we want a {\mathbb{Q}}-structure on {\mathfrak{g} = \mathfrak{u}^-\oplus \mathfrak{s}\oplus\mathfrak{a} \oplus\mathfrak{u}}, it remains to put a {\mathbb{Q}}-structure on {\mathfrak{a}}. In general this is not difficult: for instance, we can take

\displaystyle \mathfrak{a}_{\mathbb{Q}} = \left\{ \left( \begin{array}{cc} \mu I & 0 \\ 0 & -\mu I\end{array}\right):\mu\in\mathbb{Q} \right\}

in the context of the example {G=SL(2p,\mathbb{R})} and {U = \left\{\left(\begin{array}{cc} I & B \\ 0 & I\end{array}\right)\in G: B\in M(p,\mathbb{R})\right\}}.

Once we understood the constraints on a {\mathbb{Q}}-form of {G} containing {\Gamma}, we can work backwards and set

\displaystyle \mathfrak{g}_\mathbb{Q} = \mathfrak{u}^-_\mathbb{Q}\oplus \mathfrak{s}_\mathbb{Q}\oplus\mathfrak{a}_\mathbb{Q} \oplus\mathfrak{u}_\mathbb{Q}

where {\mathfrak{u}_\mathbb{Q}}, {\mathfrak{u}^-_\mathbb{Q}}, {\mathfrak{s}_\mathbb{Q}} and {\mathfrak{a}_\mathbb{Q}} are the {\mathbb{Q}}-structures from the previous paragraphs.

At this point, the proof of Theorem 4 is complete once we show the following facts:

  • {\mathfrak{g}_{\mathbb{Q}}} doesn’t depend on the choices (of {U^-}, etc.);
  • {\textrm{Ad}(\gamma) \mathfrak{g}_{\mathbb{Q}}\subset \mathfrak{g}_{\mathbb{Q}}} for all {\gamma\in\Gamma};
  • {\mathfrak{g}_{\mathbb{Q}}} is a Lie algebra.

The proof of these statements is described in the proof of Proposition 4.11 of Benoist–Miquel paper. Closing this post, let us just make some comments on the independence of {\mathfrak{g}_{\mathbb{Q}}} on the choices. For this sake, suppose that {U'} is another choice of horospherical subgroup. Denote by {\mathfrak{u}'} its Lie algebra, and let {P'=N_G(U')} be the associated parabolic subgroup. Our task is to verify that the Lie algebra {\mathfrak{p}' = \textrm{Lie}(P')} is defined over {\mathbb{Q}}. In this direction, the basic strategy is to reduce to the case when {\mathfrak{u}'} is opposite to {\mathfrak{u}} and {\mathfrak{u}^-} in order to get {\mathfrak{p}' = (\mathfrak{p}'\cap\mathfrak{p})\oplus(\mathfrak{p}'\cap\mathfrak{p}^-)}. Finally, during the implementation of this strategy, one relies on the following properties discussed in Lemma 4.8 of Benoist–Miquel article of the action of the unimodular normalizers {Q=\{g\in P: \textrm{det}_{\mathfrak{u}}\textrm{Ad}(g) = 1\}} of horospherical subgroups {U\subset P} on the space {X=G/\Gamma} with basepoint {x_0=\Gamma/\Gamma}:

  • If {Ux_0} is compact, then {Qx_0} is closed;
  • If {Qx_0} and {Q^-x_0} are closed, then {Sx_0 = (Q\cap Q^-) x_0} is closed;
  • If {Ux_0} is compact and {S x_0} is closed, then {(\Gamma\cap S)(\Gamma\cap U)} has finite index in {\Gamma\cap SU = \Gamma\cap Q}.

In any event, this completes our discussion of Theorem 4. In particular, we gave a (sketch of) proof of Theorem 1 when {U} is commutative and reflexive, and {S} is non-compact (cf. Remark 4).

Next time, we will establish Theorem 1 in the remaining cases of {U} and {S}.


As it was announced in the end of the first post of this series, we will discuss today the first half of the proof of the following result:

Theorem 1 Let {G:=SL(2p,\mathbb{R})}, {p\geq 2}, and {U := \left\{\left(\begin{array}{cc} I & B \\ 0 & I\end{array}\right)\in G: B\in M(p,\mathbb{R})\right\}}. Suppose that {\Gamma} is a discrete and Zariski dense subgroup of {G} such that {\Gamma\cap U} is cocompact. Then, {\Gamma} is commensurable to some {\mathbb{Z}}-form {G_{\mathbb{Z}}} of {G}.

Remark 1 This statement is originally due to Hee Oh, but the proof below is a particular case of Benoist–Miquel’s arguments. In particular, our subsequent discussions can be generalized to obtain the statement of Theorem 1 of the previous post in full generality.

Remark 2 Theorem 1 is not true without the higher rank assumption {p\geq 2} (i.e., {\textrm{rank}_{\mathbb{R}} G = 2p-1\geq 2}): indeed, {\Gamma = \left\langle\left(\begin{array}{cc} 1 & 3 \\ 0 & 1\end{array}\right), \left(\begin{array}{cc} 1 & 0 \\ 3 & 1\end{array}\right)\right\rangle} has infinite index in {SL(2,\mathbb{Z})}.

Our task is to construct a {\mathbb{Z}}-form {G_{\mathbb{Z}}} satisfying the conclusions of Theorem 1. This is not very easy because it must cover all possible cases of {\mathbb{Z}}-forms such as:

Example 1

  • {G_{\mathbb{Z}} = SL(2p,\mathbb{Z})};
  • {G_{\mathbb{Z}} = SL(2s,D_{\mathbb{Z}})} where {D_{\mathbb{Z}}} are integers in a division algebra over {\mathbb{Q}};
  • {G_{\mathbb{Z}} = SU(2p,\mathbb{Z}[\sqrt{2}])}”, i.e., {G_{\mathbb{Z}} = \left\{g\in SL(2p,\mathbb{Z}[\sqrt{2}]): g^{\sigma} = {}^Tg^{-1}\right\}} where {\sigma} is Galois conjugation (and {{}^Tg} is the transpose of {g}).

Before trying to construct adequate {\mathbb{Z}}-forms, let us make some preliminary reductions.

We denote by {P=N_G(U)} the parabolic subgroup normalizing of {U} in {G}: more concretely,

\displaystyle P = \left\{g\in G: g=\left(\begin{array}{cc} A & B \\ 0 & D\end{array}\right) \right\} = \{g\in G: g(\mathbb{W}) = \mathbb{W}\}

where {\mathbb{W}:=\mathbb{R}^p\times\{0\}\subset\mathbb{R}^{2p}}.

Next, we consider {U^- = \left\{\left(\begin{array}{cc} I & 0 \\ C & I\end{array}\right)\in G\right\}}. In the literature, {U^-} is called an opposite horospherical subgroup to {U}.

Since {\Gamma} is Zariski dense in {G}, there exists {\gamma_0 = \left(\begin{array}{cc} A_0 & B_0 \\ C_0 & D_0\end{array}\right)\in \Gamma} such that {\gamma_0(\mathbb{W})\oplus \mathbb{W} = \mathbb{R}^{2p}} (i.e., {\textrm{det}(C_0)\neq 0}). By taking a basis of {\mathbb{R}^{2p}} such that {\gamma_0(\mathbb{W}) = \{0\}\times\mathbb{R}^{p}}, we have that

\displaystyle \gamma_0 U\gamma_0^{-1} = U^- \ \ \ \ \ (1)

In particular, {U^-\cap\Gamma} is cocompact in {U^-} (thanks to the assumptions of Theorem 1).

Remark 3 This is the one of the few places in Benoist–Miquel argument where the Zariski denseness of {\Gamma} is used.

Remark 4 In general, the argument above works when {U} is reflexive, that is, {U} is conjugated to an opposite horospherical subgroup {U^-}.

We denote by {P^-=N_G(U^-) = \left\{g\in G: g=\left(\begin{array}{cc} A & 0 \\ C & D\end{array}\right)\in G \right\}} an opposite parabolic subgroup, and

\displaystyle L=P\cap P^- = \left\{g\in G: g=\left(\begin{array}{cc} A & 0 \\ 0 & D\end{array}\right)\in G \right\}

the common Levi subgroup of {P} and {P^-}. In particular, we have decompositions (in semi-direct products)

\displaystyle P=L\,U \quad \textrm{and} \quad P^- = L\,U^-

Let {\mathfrak{u}=\textrm{Lie}(U)} and {\mathfrak{u}^- = \textrm{Lie}(U^-)} be the Lie algebras of {U} and {U^-}. Note that {\Lambda:=\log(\Gamma\cap U)} and {\Lambda^-:=\log(\Gamma\cap U^-)} (resp.) are lattices in {\mathfrak{u}} and {\mathfrak{u}^-} (resp.). In other terms,

\displaystyle \Lambda\in X_{\mathfrak{u}} \textrm{ and } \Lambda^-\in X_{\mathfrak{u}^-}

where {X_{\ast}} is the space of lattices in {\ast\in\{\mathfrak{u}, \mathfrak{u}^-\}}.

Remark 5 Note that {\log g = g-\textrm{Id}_{2p\times 2p}} for all {g\in U} in the context of the example {G=SL(2p,\mathbb{R})} and {U := \left\{\left(\begin{array}{cc} I & B \\ 0 & I\end{array}\right)\in G: B\in M(p,\mathbb{R})\right\}}.

Observe that {L} is the intersection of the normalizers of {U} and {U^-}. Therefore, {L} acts on the spaces of lattices {X_{\mathfrak{u}}} and {X_{\mathfrak{u}^-}} via the adjoint map {\textrm{Ad}_L} (i.e., by conjugation).

As it turns out, the key step towards the proof of Theorem 1 consists in showing that the {\textrm{Ad}_{L}}-orbits of {\Lambda\in X_{\mathfrak{u}}} and {(\Lambda, \Lambda^-)\in X_{\mathfrak{u}}\times X_{\mathfrak{u}^-}} are closed. In other terms, the proof of Theorem 1 can be divided into two parts:

  • closedness of the {\textrm{Ad}_L}-orbits of {\Lambda\in X_{\mathfrak{u}}} and {(\Lambda, \Lambda^-)\in X_{\mathfrak{u}}\times X_{\mathfrak{u}^-}};
  • construction of the {\mathbb{Z}}-form {G_{\mathbb{Z}}} based on the closedness of the {\textrm{Ad}_L}-orbits above.

In the remainder of this post, we shall establish the closedness of relevant {\textrm{Ad}_L}-orbits. Then, the next post of this series will be dedicated to obtain an adequate {\mathbb{Z}}-form {G_{\mathbb{Z}}} (i.e., arithmeticity) from this closedness property.

Remark 6 Hee Oh’s original argument used Ratner’s theory for the semi-simple part of {L} to derive the desired closedness property. The drawback of this strategy is the fact that it doesn’t allow to treat some cases (such as  {G=SO(2,m)}), and, for this reason, Benoist and Miquel are forced to proceed along the lines below.

1. Closedness of the {\textrm{Ad}_L}-orbit of {\Lambda}

Consider Bruhat’s decomposition {\mathfrak{g} = \mathfrak{u}\oplus\mathfrak{l}\oplus\mathfrak{u}^-} (where {\mathfrak{l}=\textrm{Lie}(L)} is the Lie algebra of {L}) and the corresponding projection {\pi:\mathfrak{g}\rightarrow\mathfrak{u}}.

Given {g\in G}, set {M(g)=\pi \circ \textrm{Ad}_g \circ \pi\in \textrm{End}(\mathfrak{u})}, i.e.,

\displaystyle \textrm{Ad}_g = \left(\begin{array}{ccc} M(g) & \ast & \ast \\ \ast & \ast & \ast \\ \ast & \ast & \ast\end{array}\right),

and consider the Zariski open set

\displaystyle \Omega = U^- \, P = \left\{\left(\begin{array}{cc} A & B \\ C & D \end{array}\right)\in G: \textrm{det} A\neq 0\right\}.

Our first step towards the closedness of the {\textrm{Ad}_L}-orbit of {\Lambda} is to exploit the discreteness of {\Gamma} and the commutativity of {U} in order to get that the actions of the matrices {M(g)}, {g\in \Gamma\cap\Omega}, on the vectors {X} of the lattice {\Lambda} do not produce arbitrarily short vectors:

Proposition 2 The set {\{M(g) X: g\in\Gamma\cap\Omega, X\in\Lambda\}} is closed and discrete in {\mathfrak{u}}.

Proof: Let {g_n = v_n \ell_n u_n\in\Gamma} with {v_n\in U^-}, {\ell_n\in L}, {u_n\in U}, and {X_n\in \Lambda} such that {X_n' :=M(g_n)X_n \rightarrow X_{\infty}'\in \mathfrak{u}}.

Our task is to show that {X_{\infty}' = X_n'} for all {n} sufficiently large.

For this sake, note that a direct calculation reveals that {\textrm{Ad}_u\circ \pi = \pi = \pi\circ \textrm{Ad}_v} for all {u\in U} and {v\in U^-}. In particular, {X_n' = M(g_n) X_n= \textrm{Ad}(\ell_n)X_n}. Now, we use the cocompactness of {\Gamma\cap U^-} to write {v_n = \delta_n^{-1} v_n'} with {\delta_n\in \Gamma\cap U^-} and {v_n'\rightarrow v_{\infty}'} (modulo taking subsequences).

By definition, {\gamma_n:= \delta_n g_n \exp(X_n) g_n^{-1}\delta_n^{-1} = v_n'\ell_n u_n \exp(X_n) u_n^{-1} \ell_n^{-1} v_n'^{-1}\in \Gamma}. Since {U} is commutative, {u_n\exp(X_n)u_n^{-1} = \exp(X_n)} and hence

\displaystyle \Gamma\ni \gamma_n = v_n' \ell_n\exp(X_n)\ell_n^{-1} v_n'^{-1} = v_n' \exp(X_n') v_n'^{-1} \rightarrow v_{\infty}' \exp(X_{\infty}') v_{\infty}'^{-1}

as {n\rightarrow \infty}. Because {\Gamma} is discrete, it follows that {\textrm{Ad}_{v_n'}(X_n') = \textrm{Ad}_{v_{\infty}'}(X_{\infty}')} for all {n} sufficiently large. Therefore, {X_n'=\pi\circ \textrm{Ad}_{v_n'}(X_n') = \pi\circ\textrm{Ad}_{v_{\infty}'}(X_{\infty}') = X_{\infty}'} for all {n} sufficiently large. This completes the proof of the proposition. \Box

Remark 7 The fact that {U} is commutative plays a key role in the proof of this proposition.

Next, we shall combine this proposition with Mahler’s compactness criterion to study the set of determinants of the matrices {M(g)} for {g\in\Gamma}.

Proposition 3 Let {\Phi(g) = \textrm{det}_{\mathfrak{u}} M(g) = \textrm{det}(A)^{2p}} for {g=\left(\begin{array}{cc} A & B \\ C & D \end{array}\right)\in G}. Then, the set {\{\Phi(g): g\in \Gamma\}} is closed and discrete in {\mathbb{R}}.

Proof: Given {g_n\in\Gamma} such that {\Phi(g_n)\rightarrow\Phi_{\infty}}, we want to show that {\Phi(g_n) = \Phi_{\infty}} for all {n} sufficiently large.

By contradiction, let us assume that this is not the case. In particular, there is a subsequence {g_{n_k}} with {\Phi(g_{n_k})\neq 0}, i.e., {g_{n_k}\in\Gamma\cap\Omega}, and also {\Phi(g_{n_k})\neq \Phi_{\infty}} for all {k}. Note that, by definition, {\Phi(g_{n_k})} is the covolume of {M(g_{n_k})\Lambda}.

By Proposition 2, {M(g_{n_k})\Lambda} doesn’t have small vectors. Since these lattices also have bounded covolumes (because {\Phi(g_{n_k})\rightarrow\Phi_{\infty}}), we can invoke Mahler’s compactness criterion to extract a subsequence {g_{n_c}} such that {M(g_{n_c})\Lambda\rightarrow \Lambda_{\infty}}. In this setting, Proposition 2says that we must have {M(g_{n_c})\Lambda = \Lambda_{\infty}} for all {c} sufficiently large, so that {\Phi(g_{n_c}) = \Phi_{\infty}} for all {c} sufficiently large, a contradiction. \Box

Now, we shall modify {\Phi} to obtain a polynomial function {F} on {\mathfrak{u}}. For this sake, we recall the element {\gamma_0 = \left(\begin{array}{cc} A_0 & B_0 \\ C_0 & D_0 \end{array}\right)\in \Gamma} introduced in (1) (conjugating {U} and {U^-}). In this context,

\displaystyle F(X) := \Phi(\exp(X)\gamma_0) = (\textrm{det}(C_0))^{2p} (\textrm{det}(B))^{2p}

is a polynomial function of {X= \left(\begin{array}{cc} 0 & B \\ 0 & 0 \end{array}\right)\in \mathfrak{u}}. Moreover, an immediate consequence of Proposition 3 is:

Corollary 4 {F(\Lambda)} is a closed and discrete subset of {\mathbb{R}}.

The polynomial {F} is relevant to our discussion because it is intimately connected to the action of {\textrm{Ad}_L}:

Remark 8

  • on one hand, a straighforward calculation reveals that {F\circ \textrm{Ad}_{\ell}} is proportional to {F} for all {\ell\in L} (i.e., {F\circ \textrm{Ad}_{\ell} = \lambda_{\ell} F} for some [explicit] {\lambda_{\ell}\in \mathbb{R}}): see Lemma 3.12 of Benoist–Miquel paper;
  • on the other hand, some purely algebraic considerations show that {\textrm{Ad}_L} is the virtual stabilizer of the proportionality class of {F}, i.e., {\textrm{Ad}_L} is a finite-index subgroup of {\{\varphi\in \textrm{Aut}(\mathfrak{u}): F\circ \varphi \textrm{ is proportional to } F\}}: see Proposition 3.13 of Benoist–Miquel paper.

At this stage, we are ready to establish the closedness of the {\textrm{Ad}_L}-orbit of {\Lambda}:

Theorem 5 The {\textrm{Ad}_L}-orbit of {\Lambda} is closed in {X_{\mathfrak{u}}}.

Proof: Given {\ell_n\in L} such that {\textrm{Ad}_{\ell_n}\Lambda\rightarrow\Lambda_{\infty}}, we write {\textrm{Ad}_{\ell_n}\Lambda = \varphi_n(\Lambda_{\infty})} with {\varphi_n\in \textrm{Aut}(\mathfrak{u})} converging to the identity element {e\in\textrm{Aut}(\mathfrak{u})}.

Our task is reduce to show that {\varphi_{n}\in\textrm{Ad}_L} for all {n} sufficiently large. For this sake, it suffices to find {\varphi_{n_k}} stabilizing the proportionality class of the polynomial {F} (thanks to Remark 8).

In this direction, we take {X\in\Lambda_{\infty}}. By definition, {\varphi_n(X)\in \textrm{Ad}_{\ell_n}\Lambda}. Also, by Remark 8, we know that {F(\textrm{Ad}_{\ell_n}\Lambda) = \lambda_{\ell_n} F(\Lambda)} for some constant {\lambda_{\ell_n}} depending on the covolume {\Phi(\ell_n)} of {\textrm{Ad}_{\ell_n}\Lambda}. In particular, {F(\varphi_n X)\in\lambda_{\ell_n} F(\Lambda)}.

As it turns out, since the lattices {\textrm{Ad}_{\ell_n}\Lambda} converge to {\Lambda_{\infty}}, one can check that the quantities {\lambda_{\ell_n}} converge to some {\lambda_{\infty}\neq 0} related to the covolume of {\Lambda_{\infty}}. Moreover, {\lambda_{\ell_n} F(\Lambda)\ni F(\varphi_n X)\rightarrow F(X)} because {\varphi_n\rightarrow e}. Hence, we can apply Corollary 4 to deduce that {F(X)\in\lambda_{\infty} F(\Lambda)} and

\displaystyle F(\varphi_n X) = (\lambda_{\infty}/\lambda_{\ell_n}) F(X)

for all {n} sufficiently large depending on {X\in\Lambda}, say {n\geq n(X)}.

At this point, we observe that the degrees of the polynomials {F\circ\varphi_n - (\lambda_{\infty}/\lambda_{\ell_n}) F} are uniformly bounded and {\Lambda_{\infty}} is Zariski-dense in {\mathfrak{u}}. Thus, we can choose {n_0\in\mathbb{N}} such that

\displaystyle F(\varphi_n X) = (\lambda_{\infty}/\lambda_{\ell_n}) F(X)

for all {n\geq n_0} and {X\in\mathfrak{u}}.

In other terms, {\varphi_n\in\textrm{Aut}(\mathfrak{u})} stabilizes the proportionality class of {F} for all {n\geq n_0}. It follows from Remark 8 that {\varphi_n\in\textrm{Ad}_L} for all {n} sufficiently large. This completes the argument. \Box

2. Closedness of the {\textrm{Ad}_L}-orbit of {(\Lambda, \Lambda^-)}

The proof of the fact that the {\textrm{Ad}_L}-orbit of {(\Lambda,\Lambda')} is closed in {X_{\mathfrak{u}} \times X_{\mathfrak{u}^-}} follows the same ideas from the previous section: one introduces the polynomial {G(X,Y)=\Phi(\exp(X)\exp(Y))} for {(X,Y)\in \mathfrak{u}\times \mathfrak{u}^-}, one shows that {G(\Lambda\times\Lambda^-)} is closed and discrete in {\mathbb{R}}, and one exploits this information to get the desired conclusion.

In particular, our discussion of the first half of the proof of Theorem 1 is complete. Next time, we will see how this information can be used to derive the arithmeticity of {\Gamma}. We end this post with the following remark:

Remark 9 Roughly speaking, we covered Section 3 of Benoist–Miquel article (and the reader is invited to consult it for more details about all results mentioned above).Finally, a closer inspection of the arguments shows that the statements are true in greater generality provided {U} is reflexive and commutative (cf. Remarks 4 and 7).

Last week, Jon ChaikaJing Tao and I co-organized the Summer School on Teichmüller Theory and its Connections to Geometry, Topology and Dynamics at Fields Institute.

This activity was part of the Thematic Program on Teichmüller Theory and its Connections to Geometry, Topology and Dynamics, and it consisted of four excellent minicourses by Yves BenoistHee OhGiulio Tiozzo and Alex Wright.

These minicourses were fully recorded and the corresponding videos will be available at Fields Institute video archive in the near future.

Meanwhile, I decided to transcript my notes of Benoist’s minicourse in a series of four posts (corresponding to the four lectures delivered by him).

Today, we shall begin this series by discussing the statement of the main result of Benoist’s minicourse, namely:

Theorem 1 (Oh, Benoist–Miquel) Let {G} be a semisimple algebraic Lie group of real rank {\textrm{rank}_{\mathbb{R}}(G)\geq 2}. Suppose that {U} is a horospherical subgroup of {G}, and assume that {\Gamma} is a Zariski dense and irreducible subgroup of {G} such that {U\cap \Gamma} is cocompact. Then, there exists an arithmetic subgroup {G_{\mathbb{Z}}} such that {\Gamma} and {G_{\mathbb{Z}}} are commensurable.

The basic reference for the proof of this theorem (conjectured by Margulis) is the original article by Benoist and Miquel. This theorem completes the discussion in Hee Oh’s thesis where she dealt with many families of examples of semisimple Lie groups {G} (as Hee Oh kindly pointed out to me, the reader can find more details about her contributions to Theorem 1 in these articles here).

Remark 1 I came across Benoist–Miquel theorem during my attempts to understand a question by Sarnak about the nature of Kontsevich–Zorich monodromies. In particular, I’m thankful to Yves Benoist for explaining in his minicourse the proof of a result that Pascal Hubert and I used as a black box in our recent preprint here.

Below the fold, the reader will find my notes of the first lecture of Benoist’s minicourse (whose goal was simply to discuss several keywords in the statement of Theorem 1).

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During the preparation of my joint articles with K. Burns, H. Masur and A. Wilkinson about the rates of mixing of the Weil-Peterson geodesic flow (on moduli spaces of Riemann surfaces), we exchanged some emails with S. Wolpert about the sectional curvatures of the Weil-Petersson metric near the boundary of moduli spaces.

As it turns out, Wolpert communicated to us an interesting mechanism to show that some sectional curvatures can be exponentially small in terms of the square of the distance to the boundary.

On the other hand, this mechanism does not seem to be well-known: indeed, I was asked in many occasions about the behavior of the Weil-Petersson sectional curvatures near the boundary, and each time my colleagues were surprised by Wolpert’s examples.

In this short post, I will try to describe Wolpert’s construction of tiny Weil-Petersson sectional curvatures. (Of course, all mistakes below are my responsibility.)

1. Weil-Petersson metric

Recall that the cotangent bundle to the moduli space of Riemann surfaces is naturally identified with the space of quadratic differentials on Riemann surfaces.

The Weil-Petersson inner product {\langle\phi,\psi\rangle_{WP}} between two quadratic differentials {\phi} and {\psi} on a Riemann surface {S} is

\displaystyle \langle\phi,\psi\rangle_{WP} = \int_S \phi\overline{\psi}(ds^2)^{-1} \ \ \ \ \ (1)

where {ds^2} is the hyperbolic metric of {S}.

Remark 1 The quadratic differentials {\phi} and {\psi} are locally given by {\phi=f(z)dz^2} and {\psi=g(z)dz^2}, so that {\phi\psi = f(z) \overline{g(z)} dz^2d\overline{z}^2 = f(z) \overline{g(z)} |dz|^4}. In particular, we use the hyperbolic metric to obtain a {L^2}-type formula (because the area form is {|dz|^2}).

The incomplete, smooth, Kähler, negatively curved Riemannian metrics on moduli spaces of Riemann surfaces induce by the Weil-Petersson inner products are the so-called Weil-Petersson (WP) metrics.

Recall that the moduli spaces of Riemann surfaces are not compact because a hyperbolic closed geodesic {\alpha} on a Riemann surface {S} might have arbitrarily small hyperbolic length {\ell_S(\alpha)}. Moreover, the Weil-Petersson metric is incomplete because we can pinch off a hyperbolic closed geodesic {\alpha} on {S} in finite time {\leq \ell_S(\alpha)^{1/2}}. Nevertheless, the natural compactification of the moduli space of Riemann surfaces with respect to the Weil-Petersson metric turns out to be the Deligne-Mumford compactification where one adds a boundary by including stable nodal Riemann surfaces into the picture. (See Burns-Masur-Wilkinson paper and the references therein for more details.)

Today, we are interested on the order of magnitude of the Weil-Petersson sectional curvatures at a point {X} of moduli space of Riemann surfaces. More concretely, we want to understand WP sectional curvatures {K} of cotangent planes to {X} in terms of the distance {d} of {X} to the boundary (of Deligne-Mumford compactification).

By Wolpert’s work, we know that {-K=O(1/d)}, i.e., WP sectional curvatures {K} are bounded away from {-\infty} by a polynomial function of the inverse {1/d} of the distance to the boundary.

On the other hand, a potential cancellation in Wolpert’s formulas for WP curvatures makes it hard to infer upper bounds on WP sectional curvatures in terms of {1/d}. (Nevertheless, the situation is better understood for holomorphic sectional curvatures and WP Ricci curvatures: see, e.g., Melrose-Zhu paper.)

In any event, Wolpert discovered that there is no chance to expect an upper bound on all WP sectional curvatures {K} at {X} in terms of a polynomial function of the distance {d} of {X} to the boundary: in fact, we will see below that Wolpert constructed examples of Riemann surfaces {X} where some WP sectional curvature {K} behaves like {-K\sim \exp(-1/d^2)}.

2. Plumbing coordinates

The geometry of a Riemann surface near the boundary of moduli space is described by the so-called plumbing construction.

Roughly speaking, if a Riemann surface {X} is close to acquire a node at a curve {\alpha}, then we can describe an annular region {A} surrounding {\alpha} using a complex parameter {t\in\mathbb{C}} with {|t|\ll 1} and two complex coordinates {z} and {w} with the following properties.

The curve {\alpha} separates the annular region {A} into two components {A_1} and {A_2}. The coordinate {z} takes {A} to {\{|t|\leq |z|\leq 1\}} in such a way {\alpha} is mapped to {\{|z|=\sqrt{|t|}\}} and {A_1} is mapped into {\{\sqrt{|t|}\leq |z|\leq1\}}. Similarly, the coordinate {w} takes {A} to {\{|t|\leq |w|\leq 1\}} in such a way {\alpha} is mapped to {\{|w|=\sqrt{|t|}\}} and {A_2} is mapped into {\{\sqrt{|t|}\leq |w|\leq1\}}. Furthermore, we recover the annular region {A} by identifying points via the relation

\displaystyle zw=t

In the figure below, we depicted a Riemann surface {X_t} obtained from this plumbing construction near a curve separating it into two torii.

Remark 2 In the plumbing construction, the size {|t|} of the parameter {t} gives a bound on the distance of {X} to the boundary of moduli space: indeed, the hyperbolic length of the geodesic representative of {\alpha} is {\sim 1/\log(1/|t|)}. Of course, this is coherent with the idea that {zw=0} describes a node.Also, the phase of {t} is related to the so-called twist parameters.

3. Tiny Weil-Petersson curvature

Consider the plumbing construction in the figure above. It illustrates a curve {\alpha_t} separating a Riemann surface {X_t} of genus {2} into two torii {T_1} and {T_2} with natural coordinates {z} and {w}. In these coordinates, the curve {\alpha_t} is {\{z:|z|=|t|^{1/2}\} = \{w:|w|=|t|^{1/2}\}}.

We start with the quadratic differential {\psi_1=dz^2} on {T_1}. If we want to extend {\psi_2} to {T_2} and, a fortiori, {X_t}, then we need to understand the behavior of {\psi_1} in the portion {\{|t|^{1/2}\leq |w|\leq 1\}} of {T_2} intersecting the annular region surrounding {\alpha_t}. In other words, we have to describe {\psi_1} in {w}-coordinates.

For this sake, we recall that the definition of plumbing construction says that {zw=t}. Thus, {z=t/w} and the formula {dz^2 = t^2 w^{-4} dw^2} allows us to extend {\psi_1} to {X_t}.

This description has the following interesting consequence: while the Weil-Petersson size of {\psi_1} on {T_1} is {\sim 1} (because {\psi_1|_{T_1}=dz^2}), the Weil-Petersson size (cf. (1)) of {\psi_1} on {T_2} is {\sim |t|} because

\displaystyle \int_{|t|^{1/2}\leq |w|\leq 1} \underbrace{\frac{|t|^4}{|w|^8}\frac{(dw d\overline{w})^2}{|dw|^2}}_{L^2\textrm{-norm of } \psi_1=t^2w^{-4} dw^2} \underbrace{|w|^2\log^2|w|}_{(\textrm{hyperbolic metric})^{-1}}\sim \int_{|t|^{1/2}\leq |w|\leq 1} \frac{|t|^4}{|w|^6} |dw|^2

\displaystyle \sim \int_{\sqrt{|t|}}^1 \frac{|t|^4}{r^6}r dr\sim |t|

By exchanging the roles of the subindices {1} and {2} in the previous discussion, we also get a quadratic differential {\psi_2} with Weil-Petersson size {\sim 1} on {T_2} and {\sim |t|} on {T_1}.

Remark 3 The reader is invited to consult Sections 2 and 3 of this paper of Wolpert for a more detailed discussion of quadratic differentials on Riemann surfaces coming from plumbing constructions.

At this point, Wolpert notices that the Weil-Petersson sectional curvature of the plane {P(\psi_1, \psi_2)} spanned by {\psi_1} and {\psi_2} is tiny in the following sense.

It is explained in this paper of Wolpert that the Weil-Petersson curvature is a sort of “{L^4}-norm” which in the case of {P(\psi_1,\psi_2)} correspond to simply compute the size of the product {\psi_1\psi_2}. Since {\psi_n} has size {\sim 1} on {T_n} and {\sim |t|} on {T_{3-n}} for {n\in\{1,2\}}, the product {\psi_1\psi_2} has size {\sim |t|} on {X_t=T_1\cup T_2}. In summary, the Weil-Petersson curvature of {P(\psi_1,\psi_2)} is

\displaystyle \sim-|t|

On the other hand, the geodesic representative of {\alpha_t} has hyperbolic length {\ell(t)} satisfying

\displaystyle \ell(t)\sim1/\log(1/|t|),

so that the Weil-Petersson distance of {X_t} to the boundary of moduli space is

\displaystyle \sim\ell(t)^{1/2}\sim 1/\log^{1/2}(1/|t|)

In summary, we exhibited Riemann surfaces {X} at Weil-Petersson distance {d\rightarrow 0} to the boundary of moduli space where some Weil-Petersson sectional curvature has size

\displaystyle \sim -\exp(-1/d^2)

The collection of best constants {c} for the Diophantine approximation problem of finding infinitely many rational solutions {p/q\in\mathbb{Q}} to the inequality

\displaystyle |\alpha-\frac{p}{q}|<\frac{1}{cq^2}

with {\alpha\in\mathbb{R}\setminus\mathbb{Q}} is encoded by the so-called Lagrange spectrum {L}.

In a similar vein, the Markov spectrum {M} encodes best constants for a Diophantine problem involving indefinite binary quadratic real forms.

These spectra were first studied in a systematic way by A. Markov in 1880, and, since then, their structures attracted the attention of several mathematicians (including Hurwitz, Perron, etc.).

Among the basic properties of these spectra, it is worth mentioning that {L\subset M} are closed subsets of the real line. Moreover, the works of Markov from 1880 and Hall from 1947 imply that

\displaystyle L\cap(-\infty, 3) = M\cap(-\infty, 3) = \{\sqrt{5}<\sqrt{8}<\dots\}

is a increasing sequence of quadratic surds converging to {3}, and

\displaystyle L\cap[6,\infty) = M\cap[6,\infty) = [6,\infty)

On the other hand, it took some time to decide whether {L=M}. Indeed, Freiman proved in 1968 that {M\setminus L\neq\emptyset} by exhibiting a countable (infinite) collection of isolated points in {M\setminus L}. After that, Freiman constructed in 1973 an element of {M\setminus L} which was shown to be a non-isolated point of {M\setminus L} by Flahive in 1977.

A common feature of these examples of elements in {M\setminus L} is the fact that they occur before {\sqrt{12}=3.46\dots} In 1975, Cusick conjectured that there were no elements in {M\setminus L} beyond {\sqrt{12}}.

In our preprint uploaded to arXiv a couple of days ago, Gugu and I provide the following negative answer to Cusick’s conjecture:

Theorem 1 The Hausdorff dimension of {(M\setminus L)\cap (3.7, 3.71)} is {\geq 0.53128}.

Below the fold, we give an outline of the proof of this theorem.

Remark 1 The basic reference for this post is the classical book of Cusick and Flahive.

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Last week, my friends Fernando and André uploaded to the arXiv their remarkable paper “Denseness of minimal hypersurfaces for generic metrics” joint with Kei Irie (and, in fact, Fernando sent me a copy of this article about one day before arXiv’s public announcement).

The motivation for the work of Irie–Marques–Neves is a famous conjecture of Yau on the abundance of minimal surfaces.

More precisely, Yau conjectured in 1982 that a closed Riemannian {3}-manifold contains infinitely many (smooth) closed immersed minimal surfaces. Despite all the activity around this conjecture, the existence of infinitely many embedded minimal hypersurfaces in manifolds of positive Ricci curvature {M^d} of low dimensions {3\leq d\leq 7} was only established very recently by Fernando and André.

In their remarkable paper, Irie–Marques–Neves show that Yau’s conjecture is generically true in low dimensions by establishing the following stronger statement:

Theorem 1 Let {M^d} be a closed manifold of dimension {3\leq d\leq 7}. Then, a generic Riemannian metric {g} on {M^d} has a lot of minimal hypersurfaces: the union of all of its closed (smooth) embedded minimal hypersurfaces is a dense subset of {M}.

Remark 1 The hypothesis {3\leq d\leq 7} of low dimensionality is related to the fact that area-minimizing minimal hypersurfaces in dimensions {>7} might exhibit non-trivial singular sets (as it was famously proved by Bombieri–De Giorgi–Guisti), but such a phenomenon does not occur in low dimensions for “min-max” minimal hypersurfaces thanks to the regularity theories of Almgren, Pitts and Schoen–Simon.

The remainder of this post is dedicated to the proof of this theorem and, as usual, all eventual errors/mistakes in what follows are my entire responsibility.

1. Description of the key ideas

Let {M^d} be a closed manifold and {\mathcal{M}} be the space of {C^{\infty}} Riemannian metrics on {M^d}.

Given an open subset {U\subset M^d}, let {\mathcal{M}_U} be the subset of Riemannian metrics {g\in\mathcal{M}} possessing a non-degenerateclosed (smooth) embedded minimal hypersurface passing through {U}. (Here, non-degenerate means that all Jacobi fields are trivial.)

It is possible to check that a non-degenerate closed embedded minimal hypersurface {\Sigma} in {(M,g)} is persistent: more concretely, one can use the definition of non-degeneracy and the inverse function theorem to obtain that any Riemannian metric {g'} close to {g} possesses an unique closed embedded minimal hypersurface {\Sigma'} nearby {\Sigma}. In particular, every {\mathcal{M}_U} is open.

Also, let us observe (for later use) that the non-degeneracy condition is not difficult to obtain:

Proposition 2 Let {\Sigma} be a closed (smooth) embedded minimal hypersurface in the Riemannian manifold {(M^{n+1},g)}. Then, we can perform conformal perturbations to find a sequence of metrics {(g_i)_{i\in\mathbb{N}}} converging to {g} (in {C^{\infty}}-topology) such that {\Sigma} is a non-degenerate minimal hypersurface of {(M^{n+1}, g_i)} for all sufficiently large {i\in\mathbb{N}}.

Proof: This statement is Proposition 2.3 in Irie–Marques–Neves paper and its proof goes along the following lines.

Fix a bump function {h:M\rightarrow \mathbb{R}} supported in a small neighborhood of {\Sigma} and coinciding with the square {d_g(x,\Sigma)^2} of the distance function {d_g(x,\Sigma)} nearby {\Sigma}.

The metrics {g_i=\exp(2h/i)g} are conformal to {g}, and they converge to {g} in the {C^{\infty}}-topology. Furthermore, the features of the distance function imply that {\Sigma} is a minimal hypersurface of {g_i} such that the Jacobi operator {L_{\Sigma, g_i}} acting on normal vector fields verify

\displaystyle L_{\Sigma, g_i}(X) = L_{\Sigma, g}(X) - \frac{2n}{i}X

for all {i\in\mathbb{N}}. In particular, the spectrum of {L_{\Sigma, g_i}} is derived from the spectrum of {L_{\Sigma, g}} by translation by {2n/i}. Therefore, {0} doesn’t belong to the spectrum of {L_{\Sigma, g_i}} for all {i} surfficiently large, and, hence, {\Sigma} is a non-degenerate minimal hypersurface of {(M,g_i)} for all large {i}. \Box

Coming back to Theorem 1, we affirm that our task is reduced to prove the following statement:

Theorem 3 Let {M^d} be a closed manifold of dimension {3\leq d\leq 7}. Then, for any open subset {U\subset M}, one has that {\mathcal{M}_U} is dense in {\mathcal{M}}.

In fact, assuming Theorem 3, we can easily deduce Theorem 1: if {(U_i)_{i\in\mathbb{N}}} is a countable basis of open subsets of {M^d}, then Theorem 3 ensures that {\mathcal{G}=\bigcap\limits_{i\in\mathbb{N}} \mathcal{M}_{U_i}} is a countable intersection of open and dense subsets of the Baire space {\mathcal{M}}; in other terms, {\mathcal{G}} is a residual / generic subset of {\mathcal{M}} such that any {g\in\mathcal{G}} satisfies the conclusions of Theorem 1 (thanks to the definition of {\mathcal{M}_U} and our choice of {(U_i)_{i\in\mathbb{N}}}).

Remark 2 Note that, since {\mathcal{M}} is a Baire space, it follows from Baire category theorem that {\mathcal{G}} is a dense subset of {\mathcal{M}}.

Let us now explain the proof of Theorem 3. Given a neighborhood {\mathcal{V}} of a smooth Riemannian metric {g\in\mathcal{M}} on a closed manifold {M^d}, and an open subset {U\subset M}, our goal is to show that

\displaystyle \mathcal{V}\cap\mathcal{M}_U\neq\emptyset

For this sake, we apply White’s bumpy metric theorem asserting that we can find {g'\in\mathcal{V}} such that all closed (smooth) immersed minimal hypersurfaces in {(M,g')} are non-degenerate.

If {g'\in\mathcal{M}_U}, then we are done. If {g'\notin\mathcal{M}_U}, then all closed (smooth) embedded minimal hypersurfaces in {(M,g')} avoid {U}. In this case, we can naively describe the idea of Irie–Marques–Neves to perturb {g'} to get {g''\in\mathcal{V}\cap\mathcal{M}_U} as follows:

  • we perturb {g'} only in {U} to obtain {g''\in\mathcal{V}} whose volume {\textrm{vol}(M,g'')} is strictly larger than the volume {\textrm{vol}(M,g')};
  • by the so-called Weyl law for the volume spectrum (conjectured by Gromov and recently proved by Liokumovich–Marques–Neves), the {k}widths of {g''} are strictly larger than those of {g'};
  • since {k}-widths “count” the minimal hypersurfaces, the previous item implies that new minimal hypersurfaces in {(M,g'')} were produced;
  • because {g''} coincides with {g'} outside {U}, the minimal hypersurfaces of {g''} avoiding {U} are the exactly same of {g'}; thus, the new minimal hypersurfaces in {(M,g'')} mentioned above must intersect {U}, i.e., {g''\in\mathcal{M}_U\cap\mathcal{V}}.

In the sequel, we will explain how a slight variant of this scheme completes the proof of Theorem 3.

2. Increasing the volume of Riemannian metrics

Let {g'\in\mathcal{V}\setminus\mathcal{M}_U} as above. Take {h} a non-negative smooth bump function supported in {U} such that {h(x_0)=1} for some {x_0\in U}.

Consider the family {g'(t)=(1+th)g'} of conformal deformations of {g'}. Note that {\textrm{vol}(M,g'(t))>\textrm{vol}(M,g')} for all {t>0}.

From now on, we fix once and for all {t_0>0} such that {g'(t)\in\mathcal{V}} for all {t\in[0,t_0]}.

3. Weyl law for the volume spectrum

Roughly speaking, the {k}width {\omega_k(M,g)} of a Riemannian manifold {(M^{n+1},g)} is the following min-max quantity. We consider the space {\mathcal{Z}_n} of closed hypersurfaces of {M^{n+1}}, and {k}sweepouts {\Phi:\mathbb{RP}^k\rightarrow\mathcal{Z}_n} of {M}, i.e., {\Phi} is a continuous map from the {k}-dimensional real projective space {\mathbb{RP}^k} to {\mathcal{Z}_n} which is “homologically non-trivial” and, a fortiori, {\Phi} is not a constant map.

Remark 3 Intuitively, a {k}-sweepout {\Phi} is a non-trivial way of filling {M} with {k}-parameter family of hypersurfaces (which is “similar” to the way the {1}-parameter family of curves {\{z=\textrm{constant}\}\cap S^2} fills the round {2}-sphere {S^2=\{(x,y,z)\in\mathbb{R}^3: x^2+y^2+z^2=1\}}).

If we denote by {\mathcal{P}_k} the set of {k}-sweepouts {\Phi} such that no concentration of mass occur (i.e., {\lim\limits_{r\rightarrow 0}\sup\{n\textrm{-dimensional volume of }\Phi(x)\cap B_r(p): x\in \mathbb{RP}^k, p\in M\} = 0}), then the {k}-width is morally given by

\displaystyle \omega_k(M,g) = \inf\limits_{\Phi\in\mathcal{P}_k} \sup\{n\textrm{-dimensional volume of }\Phi(x): x\in\mathbb{RP}^k\}

Remark 4 Formally speaking, the definition of {k}-width involves more general objects than the ones presented above: we construct {\mathcal{Z}_n} by replacing hypersurfaces by certain {n}-dimensional flat chains modulo two, we allow arbitrary simplicial complexes {X} in place of {\mathbb{RP}^k}, etc.: see Irie–Marques–Neves’ paper for more details and/or references.

The {k}-width {\omega_k(M,g)} varies continuously with {g} (cf. Lemma 2.1 in Irie–Marques–Neves’ paper). Moreover, it “counts” minimal hypersurfaces (cf. Proposition 2.2 in Irie–Marques–Neves’ paper): if {M^{n+1}} has dimension {3\leq n+1\leq 7}, then, for each {k\in\mathbb{N}}, there is a finite collection {\{\Sigma_1,\dots,\Sigma_N\}} of mutually disjoint, closed, smooth, embedded, minimal hypersurfaces in {(M,g)} with (stability) indices {\sum\limits_{i=1}^N\textrm{index}(\Sigma_i)\leq k} such that

\displaystyle \omega_k(M,g)=\sum\limits_{i=1}^N m_i\textrm{vol}_g(\Sigma_i) \ \ \ \ \ (1)

for some integers {m_1,\dots, m_N\in\mathbb{N}}. (Here, the stability index is the quantity of negative eigenvalues of the Jacobi operator.)

Furthermore, the asymptotic behavior of {\omega_k(M,g)} is described by Weyl’s law for the volume spectrum (conjectured by Gromov and confirmed by Liokumovich–Marques–Neves): for some universal constant {a(n)>0}, one has

\displaystyle \lim\limits_{k\rightarrow\infty} \omega_k(M,g)k^{-\frac{1}{n+1}} = a(n)\textrm{vol}(M,g)^{\frac{n}{n+1}}

In particular, coming back to the context of Section 2, Weyl’s law for the volume spectrum and the fact that {g'(t_0)} has volume strictly larger than {g'} mean that we can select {k_0\in\mathbb{N}} such that the {k_0}-width {\omega_{k_0}(M,g'(t_0))} is strictly larger than the {k_0}-width of {\omega_{k_0}(M,g')}, i.e.,

\displaystyle \omega_{k_0}(M,g'(t_0)) > \omega_{k_0}(M,g') \ \ \ \ \ (2)

4. New minimal hypersurfaces intersecting {U}

We affirm that there exists {0<t_1\leq t_0} such that {g'(t_1)} possesses a closed (smooth) embedded minimal hypersurface passing through {U}.

Otherwise, for each {0\leq t\leq t_0}, all closed (smooth) embedded minimal hypersurface in {(M,g'(t))} would avoid {U}. Since {g'(t)} coincides with {g'} outside {U} (by construction), the “counting property of {k}-widths” in equation (1) would imply that

\displaystyle \omega_{k_0}(M,g'(t))\in\mathcal{C} = \left\{\sum\limits_{j=1}^N m_j\textrm{vol}_{g'}(\Sigma_j): m_j\subset\mathbb{N} \textrm{ and } \Sigma_j \textrm{ minimal in }(M,g')\right\}

for all {t\in[0,t_0]}.

On the other hand, the fact that {g'} is bumpy (in the sense of White’s theorem) permits to conclude that the set {\mathcal{C}} is countable: indeed, a recent theorem of Sharp implies that the collection of connected, closed, smooth embedded minimal hypersurfaces in {(M,g')} with bounded index and volume is finite, so that {\mathcal{C}} is countable.

Since the {k}-width depends continuously on the metric, the countability of {\mathcal{C}} implies that the function {t\mapsto \omega_{k_0}(M,g'(t))} is constant on {[0,t_0]}. In particular, we would have

\displaystyle \omega_{k_0}(M,g'(t_0)) = \omega_{k_0}(M,g'(0)) = \omega_{k_0}(M,g'),

a contradiction with (2). So, our claim is proved.

At this point, the argument is basically complete: the metric {g'(t_1)\in\mathcal{V}} has a closed (smooth) embedded minimal hypersurface {\Sigma} passing through {U}; by Proposition 2, we can perturb {g'(t_1)} (if necessary) in order to get a metric {g''\in\mathcal{V}} such that {\Sigma} is a non-degenerate closed (smooth) embedded minimal hypersurface in {(M,g'')}, that is, {g''\in\mathcal{V}\cap\mathcal{M}_U}, as desired.

This proves Theorem 3 and, consequently, Theorem 1.

Posted by: matheuscmss | October 29, 2017

Serge Cantat’s Bourbaki seminar talk 2017

About one week ago, Serge Cantat gave a beautiful talk in Bourbaki seminar about the recent works of Brown–Fisher–Hurtado on Zimmer’s program. The video of this talk and the corresponding lecture notes are available here and here.

In this post, I will transcript my notes of this talk: as usual, all errors/mistakes are my sole responsibility.

1. Introduction

General philosophy behind Zimmer’s program: given a compact manifold {M} (say, the {3}-dimensional sphere), we would like to describe the geometrical and algebraic properties of groups {\Gamma} of finite type acting faithfully on {M}; conversely, given our favorite group {G} of finite type, we want to know the class of compact manifolds {M} on which {G} acts faithfully; in this context, Zimmer’s program proposes some answers to these problems when {\Gamma} is a lattice in a Lie group.

More precisely, let {G} be a connected Lie group with finite center whose Lie algebra {\mathfrak{g}} is semi-simple, and let {A} be a connected maximal split torus of {G}. The dimension of {A}, or equivalently, the dimension of the Lie algebra {\mathfrak{a}} of {A}, is the so-called real rank of {G}, and it is denoted by {rg(G)}. Let {\Gamma} be a lattice of {G}, i.e., a discrete subgroup such that the quotient {G/\Gamma} has finite Haar measure.

For the sake of concreteness, today we will deal exclusively with the prototypical case of {G=SL_{k+1}(\mathbb{R})} and {A\simeq (\mathbb{R}_+)^k} is subgroup of diagonal matrices in {G} with positive entries.

In this setting, Zimmer’s program offers restrictions on the dimension of compact manifolds admitting non-trivial actions of {\Gamma} by smooth diffeomorphisms. In this direction, Aaron BrownDavid Fisher and Sebastian Hurtado showed here that

Theorem 1 Let {M} be a connected compact manifold. Suppose that the lattice {\Gamma} of {G=SL_{k+1}(\mathbb{R})} is uniform (i.e., {G/\Gamma} is compact).If there exists a homomorphism {\alpha:\Gamma\rightarrow\textrm{Diff}(M)} with infinite image, then

\displaystyle \textrm{dim}(M)\geq \textrm{rg}(G)

As we are going to see below, the proof of this theorem is a beautiful blend of ideas from geometric group theory and dynamical systems.

Before describing the arguments of Brown–Fisher–Hurtado, let us make a few comments of the statement of their theorem.

Remark 1 The assumption of compactness of {M} is important: indeed, any countable group acts faithfully by biholomorphisms of a connected non-compact Riemann surface (see the footnote 1 of Cantat’s text for a short proof of this fact).

Remark 2 The hypothesis of uniformity of {\Gamma} is technical: there is some hope to treat non-uniform lattices and, in fact, Brown–Fisher–Hurtado managed to recently extend their result to the case of {\Gamma=SL_{k+1}(\mathbb{Z})}.

Remark 3 The conclusion is optimal: {G=SL_{k+1}(\mathbb{R})} (and, a fortiori, any lattice {\Gamma} of {G}) acts on the real projective space {M=\mathbb{P}^k(\mathbb{R})} by projective linear transformations. On the other hand, if one changes {G}, then the inequality {\textrm{dim}(M)\geq \textrm{rg}(G)} can be improved: for example, Brown–Fisher–Hurtado proves that {\textrm{dim}(M)\geq 2n-1} when {G} is the symplectic group {Sp(2n,\mathbb{R})} of real rank {n}.

Remark 4 Concerning the regularity of the elements of {\textrm{Diff}(M)}, even though one expects similar statements for actions by homeomorphisms, Brown–Fisher–Hurtado deals only with {C^2}-diffeomorphisms because they need to employ the so-called Pesin theory of non-uniform hyperbolicity.Nevertheless, we shall assume that {\textrm{Diff}(M)=\textrm{Diff}^{\infty}(M)} in the sequel for a technical reason explained later.

Remark 5 This theorem is obvious when {\textrm{rg}(G)=1}: indeed, a compact manifold {M} whose group of diffeomorphisms is infinite has dimension {\textrm{dim}(M)\geq 1}.Hence, we can (and do) assume without loss of generality that {\textrm{rg}(G)\geq 2} in what follows.

Remark 6 The statement of Brown–Fisher–Hurtado theorem is comparable to Margulis super-rigidity theorem providing a control on the dimension of linear representations of {\Gamma}.

Read More…

Posted by: matheuscmss | October 4, 2017

Jean-Christophe Yoccoz mathematical archives

Almost one year ago, Yoccoz family gave me the immense honour of taking care of Jean-Christophe’s mathematical archives.

My general plan is to follow the same steps by Jean-Christophe when he became the responsible for Michel Herman archives, namely, I will make available at this webpage here all unpublished texts after selecting and revising them together with Jean-Christophe’s friends.

So far, the webpage dedicated to Jean-Christophe’s archives contains only an original text (circa 1986), a latex version of this text (typed by Alain Albouy, Alain Chenchiner, and myself), and some lecture notes taken by Alain Chenchiner of a talk by Jean-Christophe on the central configurations for the planar four-body problem.

Nevertheless, I hope that this webpage will be regularly updated in the forthcoming years: indeed, Jean-Christophe’s archives takes all cabinets and some corners of an entire office, and, thus, there is more than enough material to keep his friends occupied for some time. 😀

Closing this extremely short post, let me take the opportunity to announce also that Gazette des Mathématiciens (published by the French Mathematical Society) plans to publish in April 2018 a special volume (edited by P. Berger, S. Crovisier, P. Le Calvez and myself) dedicated to several aspects of Jean-Christophe’s mathematical life.


Posted by: matheuscmss | August 22, 2017

HD(M\L) < 0.986927

My friend Gugu and I have just uploaded to the arXiv our paper {HD(M\setminus L) < 0.986927}. This article continues our investigations of the Hausdorff dimension {HD(M\setminus L)} of the complement of the Lagrange spectrum {L} in the Markov spectrum {M}. More precisely, we showed in a previous paper (see also this blog post here) that {HD(M\setminus L) > 0} and we prove now that {HD(M\setminus L)<1}.

The key dynamical idea to give upper bounds on {HD(M\setminus L)} is to show that any sufficiently large element {m\in M\setminus L} is realized by a sequence {\underline{\theta}\in(\mathbb{N}^*)^{\mathbb{Z}}} whose past or future dynamics lies in the gaps of an appropriate horseshoe.

Qualitately speaking, this idea is explained by the following lemma.

Lemma 1 Fix {\Lambda}horseshoe of a surface diffeomorphism {\varphi} and {f} a height function. For simplicity, let us denote the orbits of {\varphi} by {x_n:=\varphi^n(x)}. Denote by

\displaystyle M=\{\sup\limits_{n\in\mathbb{Z}}f(x_n): x_0\in\Lambda\} \quad \textrm{ and } \quad L=\{\limsup\limits_{n\rightarrow\infty}f(x_n): x_0\in\Lambda\}

the corresponding Markov and Lagrange spectra.Let {\widetilde{\Lambda}} a subhorseshoe of {\Lambda} and set

\displaystyle m(\widetilde{\Lambda}) = \max\limits_{y\in\widetilde{\Lambda}} m(y) \quad (= \max\limits_{z\in\widetilde{\Lambda}} f(z) )

where {m(a):=\sup\limits_{n\in\mathbb{Z}}f(a_n)} is the Markov value of {a}. Consider {m\in M\setminus L} such that {m>m(\widetilde{\Lambda})}, and denote by {x_0\in\Lambda} a point with

\displaystyle m=\sup\limits_{n\in\mathbb{Z}} f(x_n) = f(x_0)

Then, either {\alpha(x_0)\cap \overline{\Lambda}=\emptyset} or {\omega(x_0)\cap \overline{\Lambda}=\emptyset} (where {\alpha(x)} and {\omega(x)} denote the {\alpha} and {\omega} limit sets of the orbit of {x}).

Proof: By contradiction, suppose that {z\in\alpha(x)\cap\widetilde{\Lambda}} and {w\in\omega(x)\cap\widetilde{\Lambda}}.

Since {m\in M\setminus L} and {m>m(\widetilde{\Lambda})}, we can select {\varepsilon>0} and {N\in\mathbb{N}} such that {f(x_n)<m-\varepsilon} for all {|n|\geq N}, and {m(\widetilde{\Lambda})<m-\varepsilon}. Also, the definitions allow us to take {m_k\rightarrow-\infty} and {n_k\rightarrow\infty} such that {x_{m_k}\rightarrow z} and {x_{n_k}\rightarrow w}.

Fix {y\in\widetilde{\Lambda}} with dense orbit and consider pieces {y_{r_k}\dots y_{s_k}} of the orbit of {y} with {y_{r_k}\rightarrow w} and {y_{s_k}\rightarrow z}.

Consider the pseudo-orbits {x_0\dots x_{n_k} y_{r_k}\dots y_{s_k}x_{m_k}\dots x_0}. By the shadowing lemma, we obtain a sequence of periodic orbits accumulating {x_0} whose Markov values converge to {m}. In particular, {m\in L}, a contradiction. \Box

In simple terms, this lemma says that an element {m\in M\setminus L} with {m>m(\widetilde{\Lambda})} is associated to an orbit {(x_n)_{n\in\mathbb{Z}}} whose past dynamics (described by {\alpha(x_0)}) or future dynamics (described by {\omega(x_0)}) avoids {\widetilde{\Lambda}}. Thus, there exists {k\in\mathbb{N}} such that either the piece {(x_n)_{n\leq -k}} of past orbit or the piece {(x_n)_{n\geq k}} of future orbit avoids a neighborhood of {\widetilde{\Lambda}} in {\Lambda} (i.e., one of these pieces of orbit lives in the gaps of {\Lambda\setminus\widetilde{\Lambda}}).

Remark 1 As it turns out, this qualitative lemma is not sufficient for our purposes and, for this reason, Gugu and I end up using a quantitative version of this lemma (called Lemma 3.1) in our paper.

Once we got some constraints on the dynamics of orbits generating elements of {M\setminus L}, our strategy to estimate {HD(M\setminus L)} consists in careful choices of {\widetilde{\Lambda}} and {\Lambda}.

For the sake of exposition, let us explain how our strategy yields some bounds for {HD((M\setminus L)\cap [3.06, \sqrt{12}])}.

Perron proved that any {m\in M\cap(-\infty, \sqrt{12}]} has the form {m=m(\underline{\theta})} where {\underline{\theta}\in\{1,2\}^{\mathbb{Z}}=\Lambda}.

Consider the subhorseshoe {\widetilde{\Lambda} = \{11,22\}^{\mathbb{Z}}} (of sequences formed by concatenations of two consecutive 1’s and two consecutive 2’s).

By applying (a quantitative version of) Lemma 1 (cf. Remark 1), one concludes that if {m=m(\underline{\theta})\in (M\setminus L)\cap [3.06, \sqrt{12}]}, then the past or future dynamics of {\underline{\theta}} lives in the gaps of {\widetilde{\Lambda}}.

This means that, up to replacing {\underline{\theta}=(\theta_n)_{n\in\mathbb{Z}}} by {(\theta_{-n})_{n\in\mathbb{Z}}}, for all {n\in\mathbb{N}} sufficiently large:

  • either there is an unique extension of {\dots\theta_0\dots\theta_n} giving a sequence whose Markov value in {(M\setminus L)\cap [3.06,\sqrt{12}]};
  • or there are two continuations {\dots\theta_0\dots\theta_n1\alpha_{n+2}} and {\dots\theta_0\dots\theta_n2\beta_{n+2}} of {\dots\theta_0\dots\theta_n} so that the interval {[[0;2\beta_{n+1}], [0;1\alpha_{n+1}]]} is disjoint from the Cantor set

    \displaystyle K(\{11,22\}):=\{[0;\gamma]:\gamma\in\{11, 22\}^{\mathbb{N}}\}

    associated to {\widetilde{\Lambda} = \{11,22\}^{\mathbb{Z}}}.

(Here, {[a_0;a_1,\dots] = a_0+\frac{1}{a_1+\frac{1}{\ddots}}} denotes continued fraction expansions.)

Note that this dichotomy imposes severe restrictions on the future {(\theta_n)_{n\geq 0}} of {\underline{\theta}} because there are not many ways to build sequences associated to {(M\setminus L)\cap[3.06,\sqrt{12}]}. More precisely, we claim that at each sufficiently large step {n},

  • either we get a forced continuation {\dots\theta_0\dots\theta_n\rightarrow \dots\theta_0\dots\theta_n\theta_{n+1}};
  • or our possible continuations are {\dots\theta_0\dots\theta_n112\alpha_{n+4}} and {\dots\theta_0\dots\theta_n221\beta_{n+4}}.

Indeed, suppose that we have two possible continuations {\dots\theta_0\dots\theta_n1\alpha_{n+2}} and {\dots\theta_0\dots\theta_n2\beta_{n+2}}. If {I(a_1,\dots,a_n)} denotes the interval of numbers in {[0,1]} whose continued fraction expansion starts by {[0;a_1,\dots, a_n,\dots]}, then the intervals {I(a_1a_2)}, {(a_1,a_2)\in\{1,2\}^2} appear in the following order on the real line:

\displaystyle I(21), I(22), I(11), I(12)


  • the continuation {\dots\theta_0\dots\theta_n12\alpha_{n+3}} is not possible (otherwise {[[0;2\beta_{n+2}]], [0;12\alpha_{n+3}]]} would contain {I(11)} and, a fortiori, intersect {K(\{11,22\})};
  • the continuation {\dots\theta_0\dots\theta_n21\beta_{n+3}} is not possible (otherwise {[[0;21\beta_{n+3}]], [0;1\alpha_{n+2}]]} would contain {I(22)} and, a fortiori, intersect {K(\{11,22\})}

so that our continuations are {\dots\theta_0\dots\theta_n11\alpha_{n+3}} and {\dots\theta_0\dots\theta_n22\beta_{n+3}}. Now, we observe that the intervals {I(11a_3)} and {I(22a_3)}, {a_3\in\{1,2\}}, appear in the following order on the real line:

\displaystyle I(222), I(221), I(112), I(111)


  • the continuation {\dots\theta_0\dots\theta_n111\alpha_{n+4}} is not possible (otherwise {[[0;22\beta_{n+3}]], [0;111\alpha_{n+4}]]} would contain {I(112)} and, a fortiori, intersect {K(\{11,22\})};
  • the continuation {\dots\theta_0\dots\theta_n222\beta_{n+3}} is not possible (otherwise {[[0;222\beta_{n+4}]], [0;11\alpha_{n+3}]]} would contain {I(221)} and, a fortiori, intersect {K(\{11,22\})}

so that our claim is proved.

This claim allows us to bound the Hausdorff dimension of

\displaystyle K:=\{[\theta_0;\theta_1,\dots]: 3.06<m(\underline{\theta})<\sqrt{12} \textrm{ as above}\}

In fact, the claim says that we refine the natural cover of {K} by the intervals {I(\theta_1,\dots, \theta_n)} by replacing it by a “forced” {I(\theta_1,\dots, \theta_n,\theta_{n+1})} or by the couple of intervals

\displaystyle I(\theta_1,\dots, \theta_n, 1, 1, 2) \quad \textrm{ and } \quad I(\theta_1,\dots, \theta_n, 2, 2, 1)

Therefore, it follows from the definition of Hausdorff dimension that

\displaystyle HD(K)\leq s_0

for any parameter {0\leq s_0\leq 1} such that

\displaystyle |I(\theta_1,\dots,\theta_n,1,1,2)|^{s_0} + |I(\theta_1,\dots,\theta_n,2,2,1)|^{s_0} \leq |I(\theta_1,\dots,\theta_n)|^{s_0} \ \ \ \ \ (1)

Because we can assume that {m=m(\underline{\theta}) = [\theta_0;\theta_1,\dots]+[0;\theta_{-1},\dots]} with {[\theta_0;\theta_1,\dots]\in K} and {[0;\theta_{-1},\dots]\in C(2) := \{[0;\gamma]: \gamma\in\{1,2\}^{\mathbb{N}}\}}, our discussion so far can be summarized by following proposition:

Proposition 2 {(M\setminus L)\cap[3.06,\sqrt{12}]} is contained in the arithmetic sum

\displaystyle C(2)+K

where {HD(K)\leq s_0} for any parameter {0\leq s_0\leq 1} satisfying (1).

Since the arithmetic sum {C(2)+K} is the projection {\pi(C(2)\times K)}, {\pi(x,y)=x+y}, of the product set {C(2)\times K}, this proposition implies the following result:

Corollary 3 {HD((M\setminus L)\cap[3.06,\sqrt{12}])\leq HD(C(2))+s_0} where {0\leq s_0\leq 1} satisfies (1).

The Hausdorff dimension of {C(2)} was computed with high accuracy by Hensley among other authors: one has {HD(C(2))<0.531291}. In particular,

\displaystyle HD((M\setminus L)\cap[3.06,\sqrt{12}])<0.531291+s_0

where {s_0} verifies (1).

Closing this post, let us show that (1) holds for {s_0=0.174813} and, consequently,

\displaystyle HD((M\setminus L)\cap[3.06,\sqrt{12}])<0.706104

For this sake, recall that

\displaystyle |I(b_1,\dots,b_n)|=\frac{1}{q_n(q_n+q_{n-1})},

where {q_j} is the denominator of {[0;b_1,\dots,b_j]}.

Hence, if we set

\displaystyle g(s) := \frac{|I(a_1,\dots,a_n,1,1,2)|^s+|I(a_1,\dots,a_n,2,2,1)|^s}{|I(a_1,\dots,a_n)|^s}

then the recurrence formula {q_{j+2}=a_{j+2}q_{j+1}+q_j} implies that

\displaystyle g(s) = \left(\frac{r+1}{(3r+5)(4r+7)}\right)^s + \left(\frac{r+1}{(3r+7)(5r+12)}\right)^s

where {r=q_{n-1}/q_n\in (0,1)}.

Because {\frac{r+1}{(3r+5)(4r+7)}\leq \frac{1}{35}} and {\frac{r+1}{(3r+7)(5r+12)}<\frac{1}{81.98}} for all {0\leq r\leq 1}, we have

\displaystyle g(s)<\left(\frac{1}{35}\right)^s + \left(\frac{1}{81.98}\right)^s

This completes the argument because {\left(\frac{1}{35}\right)^{0.174813} + \left(\frac{1}{81.98}\right)^{0.174813} < 1}.

The geodesic flow on the unit cotangent bundle {SL(2,\mathbb{R})/SL(2,\mathbb{Z})} of the modular surface {\mathbb{H}^2/SL(2,\mathbb{Z})} is intimately related to the continued fraction algorithm (see e.g. this article of Series).

In this context, the entries {(a_n)_{n\in\mathbb{N}}} of the continued fraction expansion {\alpha=\frac{1}{a_1+\frac{1}{\ddots}}} of an irrational number are related to cusp excursions of typical geodesics in the modular surface (i.e., visits to regions {\{z\in\mathbb{H}: \textrm{Im}z>T\}/SL(2,\mathbb{Z})} for {T} large).

By exploiting this relationship, Vaibhav Gadre analysed cusp excursions on the modular surface to obtain a proof of the following theorem originally due to Diamond–Vaaler:

Theorem 1 For Lebesgue almost every {\alpha=\frac{1}{a_1+\frac{1}{\ddots}}\in [0,1]}, one has

\displaystyle \lim\limits_{n\rightarrow\infty}\frac{\sum\limits_{j=1}^n a_j - \max\limits_{1\leq i\leq n} a_i}{n\log n}= \frac{1}{\log 2}

Furthermore, as it is explained in Gadre’s paper here, his analysis of cusp excursions generalize to geodesic flows on complete non-compact finite-area hyperbolic surfaces and to Teichmüller geodesic flows on moduli spaces of flat surfaces.

As the reader can infer from Section 2 of Gadre’s paper, an important ingredient in his investigation of cusp excursions is the exponential mixing property for the corresponding geodesic flows.

Partly motivated by this fact, Vaibhav Gadre and I asked ourselves if the exponential mixing result for “Weil–Petersson like” geodesic flows on surfaces obtained by Burns–Masur–M.–Wilkinson could be explored to control cusp excursions of typical geodesics.

In this post, we record lower and upper bounds obtained together with V. Gadre on the depth of cusp excursions of typical “Weil–Petersson like” geodesics.

Remark 1 As usual, all errors/mistakes are my sole responsibility.

Remark 2 Our exposition follows closely Section 2 of Gadre’s paper.

1. Ergodic averages of exponentially mixing flows

Let {(g_t)_{t\in\mathbb{R}}} be a flow on {X} preserving a probability measure {\mu}.

Suppose that {g_t} has exponential decay of correlations, i.e., there are constants {C>0} and {\delta>0} such that

\displaystyle |\int_X u_1 \cdot u_2\circ g_t \, d\mu - \int_X u_1 \, d\mu \int_X u_2 \, d\mu|\leq C e^{-\delta t} \|u_1\|_{B} \|u_2\|_{B} \ \ \ \ \ (1)

for all {t\geq 0} and all “smooth” real-valued observables {u_1, u_2\in B} in a Banach space {B\subset L^1(\mu)} containing all constant functions (e.g., {B} is a Hölder or Sobolev space).

Lemma 2 Any observable {u\in B} with {\int_X u \, d\mu = 0} satisfies

\displaystyle \int_X\left(\int_0^T u(g_t x) \, dt \right)^2\,d\mu(x)\leq \frac{2C}{\delta} T \|u\|_{B}^2

Proof: We write

\displaystyle \begin{array}{rcl} \int_X\left(\int_0^T u(g_t x) \, dt \right)^2\,d\mu(x) &=& \int_X\int_0^T\int_0^T u(g_t x) u(g_s x) \, dt \, ds \, d\mu(x) \\ &=& \int_0^T\int_0^T \left(\int_X u(g_t x) u(g_s x) \, d\mu(x)\right) \, dt \, ds \end{array}

By {g_t}-invariance of {\mu}, we get

\displaystyle \int_X\left(\int_0^T u(g_t x) \, dt \right)^2\,d\mu(x) = \int_0^T\int_0^T \left(\int_X u(g_{|t-s|} x) u(x) \, d\mu(x)\right) \, dt \, ds

The exponential decay of correlations (1) implies that

\displaystyle \int_X\left(\int_0^T u(g_t x) \, dt \right)^2\,d\mu(x) \leq C\|u\|_{B}^2 \int_0^T\int_0^T e^{-\delta |t-s|} \, dt \, ds\leq \frac{2C}{\delta}T \|u\|_{B}^2

This proves the lemma. \Box

2. Effective ergodic theorem for fast mixing flows

Suppose that {(g_t)_{t\in\mathbb{R}}} is an exponentially mixing flow on {(X,\mu)} (i.e., {g_t} satisfies (1)).

Fix {1/2<\alpha<1} and denote {T_k=T_k(\alpha)=k^{2\alpha/(2\alpha-1)}}.

Theorem 3 Given {m>1}, a function {n:\mathbb{R}\rightarrow\mathbb{N}} such that {n(T)=n(T_k)} for each {T_k\leq T < T_{k+1}}, and a sequence {\{f_j\}_{j\in\mathbb{N}}\subset B} of non-negative functions, we have for {\mu}-almost every {x\in X} that

\displaystyle \frac{1}{m} T \|f_{n(T)}\|_{L^1} - 2 T^{\alpha}\|f_{n(T)}\|_{B} \leq \int_0^T f_{n(T)}(g_t x) dt \leq mT\|f_{n(T)}\|_{L^1} + 2 T^{\alpha}\|f_{n(T)}\|_B

for all {T} sufficiently large (depending on {x}).

Proof: Given {f\in B}, let {F=f-\int_X f \, d\mu\in B}. Since {\|F\|_{B}\leq 2\|f\|_{B}}, we get from Lemma 2 that

\displaystyle \int_X\left(\int_0^T F(g_t x) \, dt \right)^2\,d\mu(x)\leq \frac{2C}{\delta} T \|F\|_{B}^2\leq \frac{8C}{\delta} T \|f\|_{B}^2


\displaystyle \mu\left(\left\{x\in X: \left(\int_0^T F(g_t x) \, dt \right)^2 \geq R\right\}\right)\leq \frac{8C}{\delta} \frac{T}{R} \|f\|_{B}^2

By setting {R=T^{2\alpha}\|f\|_{B}^2}, we obtain

\displaystyle \mu\left(\left\{x\in X: \left(\int_0^T F(g_t x) \, dt \right)^2 \geq T^{2\alpha}\|f\|_{B}^2\right\}\right)\leq \frac{8C}{\delta} T^{1-2\alpha} \ \ \ \ \ (2)

Consider the sequence {\{f_j\}_{j\in\mathbb{N}}\subset B} and let {F_j:= f_j-\int_X f_j\,d\mu}. From the estimate (2) with {T=T_k} and {F=F_{n(T_k)}}, and {T=T_{k+1}} and {F=F_{n(T_k)}}, we get

\displaystyle \mu\left(\left\{x\in X: \left(\int_0^{T_k} F_{n(T_k)}(g_t x) \, dt \right)^2 \geq T_k^{2\alpha}\|f_{n(T_k)}\|_{B}^2\right\}\right)\leq \frac{8C}{\delta} T_k^{1-2\alpha} = \frac{8C}{\delta} \frac{1}{k^{2\alpha}}


\displaystyle \mu\left(\left\{x\in X: \left(\int_0^{T_{k+1}} F_{n(T_k)}(g_t x) \, dt \right)^2 \geq T_{k+1}^{2\alpha}\|f_{n(T_k)}\|_{B}^2\right\}\right)\leq \frac{8C}{\delta} \frac{1}{(k+1)^{2\alpha}}

By Borel–Cantelli lemma, the summability of the series {\sum\limits_{i=1}^{\infty}\frac{1}{i^{2\alpha}}<\infty} for {\alpha>1/2} and the previous inequalities imply that for {\mu}-almost every {x\in X}

\displaystyle \left(\int_0^{T_k} F_{n(T_k)}(g_t x) \, dt \right)^2 \leq T_k^{2\alpha}\|f_{n(T_k)}\|_{B}^2


\displaystyle \left(\int_0^{T_{k+1}} F_{n(T_k)}(g_t x) \, dt \right)^2 \leq T_{k+1}^{2\alpha}\|f_{n(T_k)}\|_{B}^2

for all {k} sufficiently large (depending on {x}).

On the other hand, the non-negativity of the functions {f_j} says that

\displaystyle \int_0^{T_k} f_{n(T_k)}(g_t x) \, dt \leq \int_0^{T} f_{n(T_k)}(g_t x) \, dt \leq \int_0^{T_{k+1}} f_{n(T_k)}(g_t x) \, dt

for all {T_k\leq T < T_{k+1}}. Hence,

\displaystyle \begin{array}{rcl} \int_0^{T_k} F_{n(T_k)}(g_t x)\, dt &=& \int_0^{T_k} f_{n(T_k)}(g_t x)\, dt - T_k\int_X f_{n(T_k)} \, d\mu \\ &\leq& \int_0^{T} f_{n(T_k)}(g_t x)\, dt - T_k\int_X f_{n(T_k)} \, d\mu \\ &\leq& \int_0^{T_{k+1}} f_{n(T_k)}(g_t x)\, dt - T_k\int_X f_{n(T_k)} \, d\mu \\ &=& \int_0^{T_{k+1}} f_{n(T_k)}(g_t x)\, dt - T_{k+1}\int_X f_{n(T_k)} \, d\mu + (T_{k+1}-T_k)\int_X f_{n(T_k)} \, d\mu \\ &=& \int_0^{T_{k+1}} F_{n(T_k)}(g_t x)\, dt + (T_{k+1}-T_k)\int_X f_{n(T_k)} \, d\mu \end{array}

for all {T_k\leq T < T_{k+1}}.

It follows from this discussion that for {\mu}-almost every {x\in X} and all {k} sufficiently large (depending on {x})

\displaystyle T_k \|f_{n(T_k)}\|_{L^1} - T_k^{\alpha}\|f_{n(T_k)}\|_B \leq \int_0^T f_{n(T_k)}(g_t x) \, dt \leq T_{k+1} \|f_{n(T_k)}\|_{L^1} + T_{k+1}^{\alpha}\|f_{n(T_k)}\|_B

whenever {T_k\leq T < T_{k+1}}. Because {\frac{T_{k+1}}{T_k} = \left(\frac{k+1}{k}\right)^{2\alpha/(2\alpha-1)}\rightarrow 1} as {k\rightarrow\infty} and {n(T)=n(T_k)} for {T_k\leq T<T_{k+1}}, given {m>1}, the previous estimate says that for {\mu}-almost every {x\in X}

\displaystyle \frac{1}{m}T \|f_{n(T)}\|_{L^1} - 2T^{\alpha}\|f_{n(T)}\|_B \leq \int_0^T f_{n(T)}(g_t x) \, dt \leq mT \|f_{n(T)}\|_{L^1} + 2T^{\alpha}\|f_{n(T)}\|_B

for all {T} sufficiently large (depending on {x} and {m>1}). This proves the theorem. \Box

3. Bounds for certain cusp excursions

Let {S} be a compact surface with finitely many punctures equipped with a negatively curved Riemannian metric which is “asymptotically modelled” by surfaces of revolutions of profiles {y=x^r}, {r>2}, near the punctures: see this paper here for details.

In this setting, it was shown by Burns, Masur, M. and Wilkinson that the geodesic flow {g_t} on {X=T^1S} is exponentially mixing with respect to the Liouville (volume) measure {\mu}, i.e., for each {0< \theta\leq 1}, the estimate (1) holds for the space {B=C^{\theta}} of {\theta}-Hölder functions.

In this section, we want to explore the exponential mixing result of Burns–Masur–M.–Wilkinson to study the cusp excursions of {g_t}.

For the sake of simplicity of exposition, we are going to assume that there is only one cusp where the metric is isometric to the surface of revolution of the profile {y=x^r} for {r>2} and {0<x\leq 2}.

Remark 3 The general case can be deduced from the arguments below after replacing the geometric facts about the surfaces of revolution of {y=x^r} (e.g., Clairaut’s relations, etc.) by their analogs in Burns–Masur–M.–Wilkinson article (e.g., quasi-Clairaut’s relation in Proposition 3.2, etc.).

Given a vector {v\in T^1 S} with base point near the cusp, let {\phi(v)} be the angle between {v} and the direction pointing straight into the cusp of the surface of revolution of {y=x^r}. Denote by {C} the collar in {S} around the cusp consisting of points whose {x}-coordinate satisfies {1/2\leq x\leq 3/2}.

3.1. Good initial positions for deep excursions

Given a parameter {R>0}, let {X_R:=\{v\in T^1C: |\phi(v)|\leq 1/R\}}. The next proposition says that any vector in {X_R} generates a geodesic making a {\frac{3}{2R^{1/r}}}deep excursion into the cusp in bounded time.

Proposition 4 If {v\in X_R}, then the base point of {g_t(v)} has {x}-coordinate {\leq \frac{3}{2R^{1/r}}} for a certain time {0\leq t\leq a} where {a=a(r)} depends only on {r}.

Proof: By Clairaut’s relation, the {x}-coordinate along {g_t(v)} satisfies

\displaystyle x(g_t(v))^r\sin\phi(g_t(v)) = x(v)^r \sin\phi(v)

for all {t} (during the cusp excursion).

Thus, the value of the {x}-coordinate along {g_t(v)} is minimized when {\phi(g_{t_0}(v))=\pi/2}: at this instant {x(g_{t_0}(v))=x(v)(\sin\phi(v))^{1/r}}. Since {v\in X_R} implies that {x(v)\leq 3/2} and {|\phi(v)|\leq 1/R}, the proof of the proposition will be complete once we can bound {t_0} by a constant {a=a(r)}. As it turns out, this fact is not hard to establish from classical facts about geodesics on surfaces of revolution: see, for example, Equation (6) in Pollicott–Weiss paper. \Box

3.2. Smooth approximations of characteristic functions

Take {b} a smooth non-negative bump function equal to {1} on {3/4\leq x\leq 4/3} and supported on {1/2\leq x\leq 3/2} such that {\|b\|_{C^1}\leq 10}. Similarly, take {q_R} a smooth non-negative bump function equal to {1} on {|\phi|\leq 1/2R} and supported on {|\phi|\leq 1/R} such that {\|\phi\|_{C^1}\leq 3R}.

The non-negative function {f_R(v):=b(x(v))\cdot q_R(\phi(v))} is a smooth approximation of the characteristic function of {X_R}:

  • {f_R} is supported on {X_R};
  • there exists a constant {d=d(r)\geq 1} depending only on {r>2} such that
    • {\frac{1}{d}\leq R\int_S f_R \, d\mu \leq d} and
    • {\|f_R\|_{C^{\theta}}\leq d R^{\theta}}.

3.3. Deep cusp excursions of typical geodesics

At this point, we are ready to use the effective ergodic theorem to show that typical geodesics perform deep cusp excursions:

Theorem 5 For {\mu}-almost every {v\in T^1 S} and for all {T} sufficiently large (depending on {v} and {r>2}), the base point of {g_t(v)} has {x}-coordinate {\leq T^{-\frac{1}{2r}+}} for a certain time {0\leq t\leq T}. (Here, {-\frac{1}{2r}+} denotes any quantity slightly larger than {-\frac{1}{2r}}.)

Proof: Fix {\frac{1}{2}<\alpha<1}, {m=2}, {\theta>0}. Let {\xi>0} be a parameter to be chosen later and consider the function {n:\mathbb{R}\rightarrow\mathbb{N}}, {n(T)=T_k^{\xi}} for {T_k\leq T < T_{k+1}} (where {T_j:=j^{2\alpha/(2\alpha-1)}}).

The effective ergodic theorem (cf. Theorem 3) applied to the functions {f_R} introduced in the previous subsection says that, for {\mu}-almost every {v\in X} and all {T} sufficiently large (depending on {v} and {r>2}),

\displaystyle \int_0^T f_{n(T)}(g_t v) \, dt\geq \frac{1}{2}T\|f_{n(T)}\|_{L^1} - 2 T^{\alpha}\|f_{n(T)}\|_{C^{\theta}}

On the other hand, by construction, {\|f_{n(T)}\|_{L^1}\geq\frac{1}{d\, T_k^{\xi}}} and {\|f_{n(T)}\|_{C^{\theta}}\leq d \,T_k^{\theta\xi}} for a certain constant {d=d(r)>1} and for all {T_k\leq T < T_{k+1}}.

It follows that, for {\mu}-almost every {v\in X} and all {T} sufficiently large,

\displaystyle \int_0^T f_{n(T)}(g_t v) \, dt\geq \frac{1}{2d}T^{1-\xi} - 2 d T^{\alpha+\theta\xi}

If {1-\xi>\alpha+\theta\xi}, i.e., {\frac{1-\alpha}{1+\theta}>\xi}, the right-hand side of this inequality is strictly positive for all {T} sufficiently large. Since the function {f_{n(T)}} is supported on {X_{T_k^{\xi}}}, we deduce that if {\frac{1-\alpha}{1+\theta}>\xi} then, for {\mu}-almost every {v\in X} and all {T} sufficiently large, {g_{t_0}(v)\in X_{T_k^{\xi}}} (where {T_k\leq T<T_{k+1}}) for some {0\leq t_0\leq T}.

By plugging this information into Proposition 4, we conclude that, if

\displaystyle \frac{1-\alpha}{1+\theta}>\xi

then, for {\mu}-almost every {v\in X} and all {T} sufficiently large, the {x}-coordinate of {g_t(v)} is {\leq \frac{3}{2T_k^{\xi/r}}\leq 2/T^{\xi/r}} for some time {0\leq t_1\leq T+a} (where {a=a(r)} is a constant).

This proves the desired theorem: indeed, we can take the parameter {\xi} arbitrarily close to {1/2} in the previous paragraph because {\frac{1-\alpha}{1+\theta}\rightarrow 1/2} as {\alpha\rightarrow 1/2} and {\theta\rightarrow 0}. \Box

3.4. Very deep cusp excursions are atypical

Closing this post, let us now show that an elementary argument à la Borel–Cantelli implies that a typical geodesic doesn’t perform very deep cusp excursions:

Theorem 6 For {\mu}-almost every {v\in T^1 S} and for all {T} sufficiently large (depending on {v} and {r>2}), the base point of {g_t(v)} has {x}-coordinate {>T^{-\frac{1}{r}-}} for all times {0\leq t\leq T}. (Here, {-\frac{1}{r}-} denotes any quantity slightly smaller than {-\frac{1}{r}}.)

Proof: Let {\xi>0} and {\beta>0} be parameters to be chosen later, and denote {T_k=k^{\beta}}.

By elementary geometrical considerations about surfaces of revolution (similar to the proof of Proposition 4), we see that if the base point of {w\in T^1S} has {x}-coordinate {x=T_k^{-\xi}}, then the base point of {g_s(v)} has {x}-coordinate in {[(1/2)T_k^{-\xi}, 2T_k^{-\xi}]} for all {|s|\sim T_k^{-\xi}}.

Therefore, if we divide {[0,T_k]} into {\sim T_k^{1+\xi}} intervals {I_j^{(k)}=[a_j^{(k)}, b_j^{(k)}]} of sizes {\sim T_k^{-\xi}}, then

\displaystyle \{v: \exists t\in I_j^{(k)} \textrm{ with } x(g_t(v)) = T_k^{-\xi} \} \subset \{v: x(g_{a_j^{(k)}}(v))\in [\frac{1}{2}T_k^{-\xi}, 2T_k^{-\xi}]\}

Since the Liouville measure {\mu} is {g_t}-invariant and the surface of revolution of the profile {y=x^r} has the property that the volume of the region {\{w\in T^1S: x(w)\in [\frac{1}{2R}, \frac{2}{R}]\}} is {O(R^{r+1})}, we deduce that

\displaystyle \mu(\{v\in T^1S: \exists t\in I_j^{(k)} \textrm{ with } x(g_t(v)) = T_k^{-\xi} \})=O(1/T_k^{\xi(r+1)})

for all {j}. Because we need {\sim T_k^{1+\xi}} indices {j} to cover the time interval {[0,T_k]}, we obtain that

\displaystyle \mu(\{v\in T^1S: \exists t\in [0,T_k] \textrm{ with } x(g_t(v)) = T_k^{-\xi} \})=O(1/T_k^{\xi r-1}) \ \ \ \ \ (3)

We want to study the set {A_k=\{v\in T^1S: \exists t\in [0,T_k] \textrm{ with } x(g_t(v)) \leq T_k^{-\xi}\}}. We divide {A_k} into {B_k:=A_k\cap\{v\in T^1S: x(v)\leq 2T_k^{-\xi}\}} and {C_k:=A_k\setminus B_k}. Because {\mu(B_k) \leq \mu(\{v\in T^1S: x(v)\leq 2T_k^{-\xi}\}) = O(1/T_k^{\xi(r+1)})}, we just need to compute {\mu(C_k)}. For this sake, we observe that

\displaystyle C_k\subset \{v: \exists t\in [0,T_k] \textrm{ with } x(g_t(v)) = T_k^{-\xi}\}

and, a fortiori, {\mu(C_k)=O(1/T_k^{\xi r-1})} thanks to (3). In particular,

\displaystyle \mu(\{v\in T^1S: \exists t\in [0,T_k] \textrm{ with } x(g_t(v)) \leq T_k^{-\xi}\})=\mu(A_k) = O(1/T_k^{\xi r-1})

Note that the series {\sum\limits_{k=1}^{\infty}1/T_k^{\xi r - 1} = \sum\limits_{k=1}^{\infty}1/k^{\beta(\xi r - 1)}} is summable when {\beta(\xi r-1)>1}, i.e., {\xi>\frac{1}{r}(1+\frac{1}{\beta})} In this context, Borel–Cantelli lemma implies that, for {\mu}-almost every {v\in T^1S}, the {x}-coordinate {g_t(v)} is {>T_k^{-\xi}} for all {t\in [0, T_k]} and all {T_k=k^{\beta}} sufficiently large (depending on {v}). Since {\frac{T_{k+1}}{T_k}\rightarrow 1} as {k\rightarrow\infty}, we conclude that if

\displaystyle \xi>\frac{1}{r}(1+\frac{1}{\beta})

then for {\mu}-almost every {v\in T^1S}, the {x}-coordinate {g_t(v)} is {>T^{-\xi}} for all {t\in [0, T]} and all {T} sufficiently large (depending on {v}).

This ends the proof of the theorem: in fact, by letting {\beta\rightarrow\infty}, we can take {\xi>1/r} arbitrarily close to {1/r} in the previous paragraph. \Box

Remark 4 By Theorems 5 and 6, a typical geodesic {\{g_t(v)\}_{t\in\mathbb{R}}} enters the region {\{w: x(w)\leq T^{-1/2r+}\}} while avoiding the region {\{w: x(w)\leq T^{-1/r-}\}} during the time interval {[0,T]} (for all {T} sufficiently large).Of course, the presence of a gap between {T^{-1/2r+}} and {T^{-1/r-}} motivates the following question: is there an optimal exponent {\frac{1}{2r}\leq \xi\leq \frac{1}{r}} such that a typical geodesic {\{g_t(v)\}_{t\in\mathbb{R}}} enters {\{w: x(w)\leq T^{-\xi+}\}} while avoiding {\{w: x(w)\leq T^{-\xi-}\}} during the time interval {[0,T]} (for all {T} sufficiently large)?

Remark 5 Contrary to the cases discussed in Gadre’s paper, we can’t show that typically there is only one “maximal” cusp excursion in our current setting (of geodesic flows on negatively curved surfaces with cusps modelled by surfaces of revolution of profiles {y=x^r}). In fact, the exponential decay of correlations estimate proved by Burns–Masur–M.–Wilkinson is not strong enough to provide good quasi-independence estimates for consecutive cusp excursions (with the same quality of Lemma 2.11 in Gadre’s paper).

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