Notes of Frederic Paulin’s lecture

Equidistribution in the Heisenberg group

Joint with J. Parkkonen.

I view the Heisenberg group as a real quadric

\displaystyle  \begin{array}{rcl}  Heis_3=\{(w_0,w)\in{\mathbb C}\times{\mathbb C}\,;\,2\Re e(w_0)=|w|^2\}. \end{array}

In these coordinates, Haar measure is {d(\Im m(w_0))\frac{dwd\bar{w}}{i}}. The map {(w_0,w)\mapsto w}, {Heis_3\rightarrow{\mathbb C}}, is a group homomorphism.

1. Equidistribution of rational points

The field {{\mathbb Q}(i)} will play the role of the rationals, and {\mathcal{O}={\mathbb Z}[i]} the role of integers. requiring that {w_0} and {w} belong to {K} defines the rational Heisenberg group {Heis_3({\mathbb Q})}. Elements of {Heis_3({\mathbb Q})} can be uniquely written as fractions {(\frac{a}{c},\frac{b}{c})} with mutually prime Gaussian integers {a}, {b}, {c\in{\mathbb Z}[i]}.

Theorem 1 At rational points of denominator of modulus {<s} (the number of them in a ball grows like {s^{-4}}), put a unit Dirac mass. Then the normalized sum weakly converges to Haar measure up to the multiplicative constant

\displaystyle  \begin{array}{rcl}  c_1=2\pi\frac{|D_k|^{3/2}\zeta_K(3)}{\zeta(3)}. \end{array}

Corollary 2 Modulo the integer points, {Heis_3({\mathbb Z})}, The numbder of rational points of denominator {<s} is {\sim \frac{1}{c_1}s^4}.

2. Counting arithmetic chains

2.1. Chains

Let {h=-z_0\bar{z}_2-z_2\bar{z}_0+|z_1|^2} be the signature {(1,2)} Hermitian form in {{\mathbb C}^3}. The Poincaré hypersphere

\displaystyle  \begin{array}{rcl}  \mathcal{HS}\{[z_0:z_1:z_2]\in P^2({\mathbb C})\,;\,h(z_0,z_1,z_2)=0\}. \end{array}

is a compactification of {Heis_3}. Indeed, {Heis_3} embeds in it via {(w_0,w)\mapsto[1:w:w_0]} as the complement of single point {\infty=[0:0:1]}.

On {Heis_3} we shall use the Cygan distance (aka Koranyi distance). It is the unique left-invariant on {Heis_3} such that

\displaystyle  \begin{array}{rcl}  d_{Cyg}((w_0,w),(0,0))=\sqrt{2|w_0|}. \end{array}

And also a modified Cygan distance

\displaystyle  \begin{array}{rcl}  d''_{Cyg}((w_0,w),(0,0))=\frac{2|w_0|}{\sqrt{|w|^2+2|w_0|}}. \end{array}

It is equivalent to {d_{Cyg}} or the a Carnot-Caratheodory metric. We need this exact expression in order to state sharp estimates.

Definition 3 (von Staudt) A chain is the intersection of Poincaré’s hypersphere with a projective line.

In {Heis_3}, those which contain {\infty} are lines {\{w=const.\}}. The others are ellipses which projet to circles in {{\mathbb C}}.

2.2. Arithmetic groups and chains

{SL(3,{\mathbb C})} on {P^2({\mathbb C})} and preserves lines. The subgroup {SU_h} preserves the Hermitian form {h} and therefore acts on Poincarés hypersphere. Let

\displaystyle  \begin{array}{rcl}  \Gamma=SU_h \cap SL_3({\mathbb Z}[i]). \end{array}

This is a discrete subgroup of {SU_h}, which preserves chains.

Definition 4 A chain is arithmetic if its stabilizer in {\Gamma} has a dense orbit in the chain.

Theorem 5 Let {C_0} be an arithmetic chain. There exist constants {\kappa>0} and {c_2>0} (depending on {C_0}) such that

\displaystyle  \begin{array}{rcl}  |Heis_3({\mathbb Z})\setminus\{c\in\Gamma C_0\,;\, \mathrm{diameter}_{d''_{Cyg}}(c)\geq\epsilon\}|\sim c_2 \epsilon^{-4}(1+O(\epsilon^\kappa)). \end{array}

3. Complex hyperbolic geometry

This enters the proof. Complex hyperbolic space has curvature between {-4} and {-1}. We use the Siegel domain model

\displaystyle  \begin{array}{rcl}  \{(w_0,w)\in{\mathbb C}\,;\, 2\Re e(w_0)-|w|^2>0\} \end{array}

and the projective model

\displaystyle  \begin{array}{rcl}  \{[z_0:z_1:z_2]\in P^2({\mathbb C})\,;\, h(z_0,z_1,z_2)<0\} \end{array}

related by the map {(w_0,w)\mapsto [1:w:w_0]}. Taking closures, we recover the previously encountered {Heis_3} and {\mathcal{HS}}. The identity component of their isometry group is {PSU_h}. The subset

\displaystyle  \begin{array}{rcl}  \{(w_0,w)\in{\mathbb C}\,;\, 2\Re e(w_0)-|w|^2>1\} \end{array}

is a horoball centered at {\infty}. {PSU_h} is transitive on horoballs.

Definition 6 A complex geodesic is the intersection of {H^2({\mathbb C})} with a projective line.

Complex geodesics are totally geodesic with curvature {-4}. Chains {c} coincide with ideal boundaries of complex geodesics {L}. A chain {c} is arithmetic if and only if {L} is, i.e. the stabilizer of {L} in {\Gamma} has finite covolume on {L}.

4. Equidistribution of common perpendiculars

Fix an arithmetic complex geodesic {L_0} with ideal boundary {C_0}. We project {H^2({\mathbb C})} radially from {\infty} onto {Heis_3} via

\displaystyle  \begin{array}{rcl}  (w_0,w)\mapsto(\frac{1}{2}|w|^2+i\Im m(w_0),w). \end{array}

This allows to identify the boundary of the model horoball {H} with {Heis_3}.

Consider common perpendiculars

  • of {H} and {H'}, horoballs.
  • of {H} and a complex geodesic.

I both cases, these are vertical segments.

Theorem 7 Put a unit Dirac mass at each footpoint {x_\gamma} of a common perpendicular of {H} with {\gamma H} whose height is {>t}. Renormalize by {e^{-4t}}. Then the resulting sum weakly converges to Haar measure on {Heis_3} up to a mltiplicative constant

\displaystyle  \begin{array}{rcl}  c_3=2\pi\frac{vol()vol()}{vol()}. \end{array}

The values of {\zeta} functions in previous theorem arive from these volumes.

4.1. Proof

Use a formula for the Poisson measure and the measure of maximal entropy in terms of the Cygan distance. Then establish exponential decay of correlations. Here is the crucial Lemma.

Lemma 8

\displaystyle  \begin{array}{rcl}  d(H,\gamma H)=2\log|c| \end{array}

if the point at infinity of {\gamma H} is {(\frac{a}{c},\frac{b}{c})}.

\displaystyle  \begin{array}{rcl}  d(H,\gamma L_0)=-\log(\frac{\mathrm{diameter}(\gamma C_0)}{\sqrt{2}}). \end{array}

whence the change of variable {s=e^t}.

Next seminar on may 14th.

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Notes of Frederic Jean’s lecture

Volume de Hausdorff en géométrie sous-riemannienne

Avec Roberta Ghezzi.

1. Volumes

{M} variété de dimension {n}, {\Delta} sous-fibré de {TM}, {d} la distance sous-riemannienne associée (Carnot-Carathéodory). On construit les sous espaces {\Delta^s} par récurrence : {\Delta^{s+1}=[\Delta,\Delta^s]}.

1.1. La mesure de Hausdorff

La {\alpha}-mesure de Hausdorff {\mathcal{H}^\alpha(A)} d’un sous-ensemble {A} est la limite, quand {\epsilon} tend vers 0, de l’inf sur tous les recouvrements de {A} de la somme des diamètres élevés à la puissance {\alpha}. Si on insiste pour recouvrir par des boules, on obtient la {\alpha}-mesure de Hausdorff sphérique {\mathcal{S}^\alpha(A)}.

1.2. Le cas équirégulier

On suppose que {\Delta} satisfait la condition de Chow-Hörmander, i.e. qu’il existe {r} tel que {\Delta^r=TM}. Dans le cas équirégulier (les espaces du drapeau forment des sous-fibrés), le théorème ball-box encadre les boules par des parallélépipèdes dans des coordonnées adaptées (on cite en général Mitchell, mais la bonne référence est Bellaïche). Il en résulte que la dimension de Hausdorff {Q} est un entier, et que {\mathcal{Q}^\alpha} a une densité bornée (et d’inverse borné) localement par rapport à tout volume lisse.

2. Le cas non équirégulier

On fait l’hypothèse que le lieu singulier {\Sigma} (lieu des points au voisinage desquels les dimensions des sous-espaces {\Delta^s} ne sont pas constantes) est de mesure nulle. Donc la dimension de Hausdorff {Q} ne peut prendre qu’un nombre fini de valeurs localement, des entiers. Donc un volume {vol_H=\mathcal{S}^Q}.

Theorem 1 La décomposition {vol_H={vol_H}_{|Reg}+{vol_H}_{|\Sigma}} est la décomposition de Lebesgue de {vol_H} par rapport à un volume lisse, i.e. {{vol_H}_{|Reg}} est absolument continue et {{vol_H}_{|\Sigma}} est singulière.

2.1. Densité bornée ?

Il peut arriver que la dimension de Hausdorff du lieu singulier soit {>} à celle du lieu régulier. On va supposer que ce n’est pas le cas. On s’intéresse à la densité {\rho} de {{vol_H}_{|Reg}} par rapport à un volume lisse. Elle est finie sur le lieu régulier. Si la dimension est localement constante, elle est strictement positive. En revanche, dès que {\Sigma\not=\emptyset}, elle n’est pas bornée.

Theorem 2 La densité {\rho} de {{vol_H}_{|Reg}} par rapport à un volume lisse (une {n}-forme {\omega}) est {\sim 1/\nu}, où, pour tout point {q}, si on note {X_1,\ldots,X_n} des champs de vecteurs engendrant {\Delta},

\displaystyle  \begin{array}{rcl}  \nu(q)=\max\{\omega(X_{I_1},\ldots,X_{I_n})(q)\,;\,\sum_i I_i =Q\} \end{array}

2.2. Densité {L^1} ?

La densité {\rho} n’est pas toujours {L^1_{loc}}. C’est le cas si et seulement si le le volume de Hausdorff des boules est fini.

Theorem 3 Condition nécessaire et suffisante, algébrique, pour que le volume de Hausdorff des boules soit fini.

2.3. Conséquences

Si le lieu singulier contient une hypersurface, alors le volume des boules est infini.

Dans le cas {C^\infty} générique,

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Notes of Cornelia Drutu’s lecture

Ranks of mapping class groups

{S} orientable genus {g} surface, {p} boundary components. Define

\displaystyle  \begin{array}{rcl}  \mathrm{Complexity}(S):=\xi(S)=3g+p-3. \end{array}

The mapping class group {MCG(S)} is the group of isotopy classes of orientation preserving homeomorphisms of {S}.

Assume complexity is at least 2 (otherwise, MCG is virtually free).

We shall use various notions of rank.

1. Version 1

Let {X} be a Hadamard manifold. Assume {X} has a discrete cocompact isometry group {G}. The quasiflat rank is the maximal dimension of quasiflats, i.e. quasi-isometrically embedded Euclidean spaces. Replacing quasi-isometrically by isometrically does not change the definition (this is a theorem of Bruce Kleiner).

Example 1 Symmetric spaces. There, quasi-flat rank is the usual rank, even for non uniform lattices.

Indeed, Lubotzky-Moses-Raghunathan show that non uniform lattices of higher rank are quasi-isometrically embedded.

This definition does not quite serve our purposes, since

Theorem 1 (Kapovich-Leeb) No geometric action of {MCG(S)} on Hadamard manifolds.

Nevertheless, MCG’s behave like {CAT(0)} groups in many respects,

Theorem 2 (Farb-Lubotzky-Minsky) {MCG(S)} has {\xi(G)}-dimensional quasi-flats.

Quasi-flat rank conjecture (Hamenstädt, Behrstock-Minsky). QFlatRank {=\xi(S)} ?

2. Version 2

Let {X} be a Hadamard manifold. A {k}-sphere is {X} in a Lipschitz map {f} of round {S^k} to {X}. So is a {k+1}-ball. Define

\displaystyle  \begin{array}{rcl}  \mathrm{Fill}(f)&=&\inf\{\mathrm{volume}(g)\,;\,g\textrm{ fills }f\},\\ Iso_k(x)&=&\sup\{\mathrm{Fill}(f)\,;\,\mathrm{volume}(f)\leq A\,x^k\}. \end{array}

Definition 3 (Filling rank)

\displaystyle  \begin{array}{rcl}  \mathrm{FillRank}=\max\{k\,;\,Iso_k(x)\leq x^{k+1}\}. \end{array}

Possible alternate definition: {\min\{k\,;\,Iso_k(x)\sim x^{k}\}}.

Proposition 4 For uniform lattices in symmetric spaces, FillRank=QFlatRank.

However, for non-uniform lattices, the story is still open. Robert Young solved the case of {Sl(n,{\mathbb Z})}.

Theorem 5 (Behrstock-Drutu) For {k\leq\xi(S)-1}, {Iso_k(x)\sim x^{k+1}}.

For {k\geq\xi(S)}, {Iso_k(x)=o(x^{k+1})}.

In small genera, we can improve the second estimate.

Difficulties: cutting spheres in pieces.

3. Version 3

Let {X} be a Hadamard manifold. Let us define higher dimensional divergence: it measures how geodesics diverge, and simultaneously, filling properties.

Fix a base point {p_0}. Let {f:S^k\rightarrow X\setminus B(p_0,x)} be a sphere of volume volume{(f)\leq A\,x^k}, outside a large ball. Define

\displaystyle  \begin{array}{rcl}  \mathrm{divergence}_k(f)&=&\inf\{\mathrm{volume}(g)\,;\,g\textrm{ fills }f\textrm{ outside }B(p_0,\lambda x)\}.\\ Div_k(x)&=&\sup\{\mathrm{divergence}_k(f)\,;\,\mathrm{volume}(f)\leq A\,x^k \textrm{ outside }B(p_0,x)\}. \end{array}

Example 2 {X} symmetric space. Then

  1. If {k<\mathrm{QFlatRank}(X)-1}, {Div_k(x)\sim x^{k+1}}.
  2. If {k=\mathrm{QFlatRank}(X)-1}, {Div_k(x)\sim e^x} (Leuzinger).
  3. If {k\geq\mathrm{QFlatRank}(X)}, {Div_k(x)\leq x^{k+1}} (Hindawi), improved by Wenger into {\sim x^{k+1}}.

Theorem 6 (Behrstock-Drutu)

  1. If {k<\xi(S)-1}, {Div_k(x)\geq x^{k+2}}.
  2. If {k\geq \xi(S)}, {Div_k(x)=o(x^{k+1})}.

Again, in low genera, one can improve a bit. Also, 1-dimensional divergence is better understood.

Theorem 7 (Abrams-Brady-Duchin-Young)

\displaystyle  \begin{array}{rcl}  Div_1(x)\leq x^4. \end{array}

4. Intermediate results

Here are two facts that hold in a larger generality.

If a group {G} is {\mathcal{F}_{\infty}} and has a combing, then

\displaystyle  \begin{array}{rcl}  Iso_k(x)\leq x^{k+1}. \end{array}

If {G} is hyperbolic, then all {Iso_k} are linear (for Lipschitz maps, this is due to Urs Lang).

5. Proof

I explain why MCG’s do not behave like Hadamard spaces.

For all {k\leq} QFlatRank {=\xi(S)}, there exists a {k}-quasflat that is maximal, i.e. not contained in a quasi-flat of higher dimension. This is different from the case of symmetric spaces. These lower dimensional quasiflats arise from multicurve splittings of the surface.

For instance, split {S=U_1\cup U_2} where {U_1} is an annulus. Then Dehn twists in {U_1} provide undistorted subgroups {{\mathbb Z}\times MCG(U_2)} in MCG{(S)}.

Our main tool: let {W\subset S} be a subsurface. Then there is a Lipschitz map of {MCG(S)} to {<pA>\times MCG(W^c)} which is locally constant outside {<pA>\times MCG(W)}.

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Notes of Michel Ledoux Orsay lecture

Comment la diffusion de la chaleur explore des inégalités fonctionnelles et géométriques

Des propriétés de monotonie de l’équation de la chaleur. Ancien, mais regain d’intér\^et motivé par l’analyse harmonique booléenne et l’informatique théorique.

1. L’équation de la chaleur dans {{\mathbb R}^n}

\displaystyle  \begin{array}{rcl}  h_t(x)=(4\pi t)^{-t/2}\exp(-|x|^2/4t) \end{array}

engendre un opérateur de convolution {H_t (f)=h_t\star f} qui forme un semi-groupe, solution de l’équation de la chaleur

\displaystyle  \begin{array}{rcl}  \partial_t u=\Delta u,\quad u(.,0)=f. \end{array}

Au moyen de la mesure gaussienne {d\gamma_n}, on peut aussi écrire

\displaystyle  \begin{array}{rcl}  H_t f(x)=\int f(x+\sqrt{2t}y)\,d\gamma_n(y), \end{array}

et, au moyen du mouvement brownien {W_t^x} issu de {x},

\displaystyle  \begin{array}{rcl}  H_t f(x)=\mathop{\mathbb E}( f(W_t^x)). \end{array}

2. L’inégalité de Hölder

Soit {\theta\in]0,1[}. L’inégalité de Hölder énonce que

\displaystyle  \begin{array}{rcl}  \int f^\theta g^{1-\theta}\leq(\int f)^\theta(\int g)^{1-\theta}. \end{array}

On va montrer que, pour tout {t>0},

\displaystyle  \begin{array}{rcl}  H_t(f^\theta g^{1-\theta})\leq (H_t f)^{\theta}(H_t g)^{1-\theta}. \end{array}

L’inégalité de Hölder en découle.

On utilise une inégalité d’interpolation, attribuée parfois à Duhamel : on montre que

\displaystyle  \begin{array}{rcl}  \Lambda(s)=(H_{t-s} f)^{\theta}(H_{t-s} g)^{1-\theta} \end{array}

est décroissante. On note {F=\log H_{t-s}f}, {G=\log H_{t-s}g} et {K=\theta F+(1-\theta)G}. On dérive.

\displaystyle  \begin{array}{rcl}  \Lambda'(s)&=&H_s(\Delta(e^K)-[\theta e^{-F}\Delta F+(1-\theta)e^{-G}\Delta G])\\ &=&H_s(|\nabla K|^2 -\theta|\nabla F|^2 -(1-\theta)|\nabla G|^2) \leq 0. \end{array}

Remarquer que gr\^ace à la chaleur, on se ramène à une inégalité quadratique, ce qui ouvre la porte à d’autres inégalités géométriques.

3. Inégalités de Brascamp-Lieb

Elles améliorent Hölder le long de certaines directions. Soient {u_1,\ldots,u_m} des vecteurs unitaires de {{\mathbb R}^n}, {0\leq c_k\leq 1}, {f_k:{\mathbb R}\rightarrow{\mathbb R}_+}. Alors

\displaystyle  \begin{array}{rcl}  \int_{{\mathbb R}^n}\prod_{k=1}^{m}f_k^{c_k}(\langle u_k,x\rangle)\,dx \leq \prod_{k=1}^{m}\int_{{\mathbb R}^n}(f_k(\langle u_k,x\rangle)\,dx)^{c_k}. \end{array}

Keith Ball en a mis en évidence une version géométrique. Supposons que

\displaystyle  \begin{array}{rcl}  \sum_{k=1}^{m}c_k u_k \otimes u_k=Id_{{\mathbb R}^n}, \end{array}

(décomposition de l’identité). Alors l’inégalité résulte de la décroissance sous le semi-groupe de la chaleur de

Rempla\c cons la mesure de Lebesgue par la mesure gaussienne. C’est la mesure invariante du semi-groupe d’Ornstein-Uhlenbeck,

\displaystyle  \begin{array}{rcl}  P_t f(x)=\int f(e^{-t}x+\sqrt{1-e^{-2t}}y)\,d\gamma_n(y), \end{array}

associé au générateur infinitésimal {L=\Delta-x\cdot\nabla}.

3.1. Exemple de décomposition de l’identité

Soit {u_1=(1,0)} et {u_2=(\rho,\sqrt{1-\rho^2})}. Si {\rho^2 c_1 c_2=(c_1-1)(c_2-1)}, on a une décomposition de l’identité, et l’inégalité de Brascamps-Lieb donne

\displaystyle  \begin{array}{rcl}  \int f_1^{c_1}P_t(f_2^{c_2})d\gamma\leq (\int f_1^{1/c_1})^{c_1}(\int f_2^{1/c_2})^{c_2}. \end{array}

Cela donne le Théorème d’hypercontractivité de Nelson (1966),

\displaystyle  \begin{array}{rcl}  \|P_t f_2\|_{p_2}\leq\|f_2\|_{p_1}, \end{array}

où on a posé {p_i=1/c_i}, soit

\displaystyle  \begin{array}{rcl}  \frac{1}{\rho^2}=\frac{p_1-1}{p_1-1}. \end{array}

3.2. Résumé

On a examiné une expression du type

\displaystyle  \begin{array}{rcl}  \int\int f(x)^\alpha g(\rho x+\sqrt{1-\rho^2}y)^\beta \,d\gamma(x)\,d\gamma(y), \end{array}

On a remplacé {f} par {P_t f}.

4. Généralisation possible

Soit {J} une fonction réelle sur un produit d’intervalles. Quant a-t-on

\displaystyle  \begin{array}{rcl}  \int\int J(f(x),g(\rho x+\sqrt{1-\rho^2}y)) \,d\gamma(x)\,d\gamma(y)\leq J(\int f\,d\gamma,\int g\,d\gamma) ? \end{array}

On montre que ceci a lieu si et seulement si {J} a la propriété de concavité suivante

\displaystyle  \begin{array}{rcl}  \begin{pmatrix} \partial_{11}J & \rho\partial_{12}J \\ \rho\partial_{21}J & \partial_{22}J \end{pmatrix}\leq 0. \end{array}

Appelons cela {\rho}-concavité.

4.1. Exemples


\displaystyle  \begin{array}{rcl}  J(u,v)=u^\alpha v^\beta. \end{array}

Mossel, Neeman (2012)

\displaystyle  \begin{array}{rcl}  J(u,v)=\mathop{\mathbb P}(X\leq\Phi^{-1}(x),Y\leq\Phi^{-1}(v)) \end{array}

{X} et {Y} sont gaussiennes standard et {\rho}-corrélées, {\Phi} la fonction de répartition gaussienne. Comme {J(0,v)=J(u,0)=0} et {J(1,1)=1}, pour tous ensembles mesurables {A} et {B},

\displaystyle  \begin{array}{rcl}  \int 1_A P_t 1_B \,d\gamma\leq J(\gamma(A),\gamma(B)). \end{array}

Si {H} et {K} sont des demi-espaces parallèles de m\^emes mesures gaussiennes que {A} et {B},

\displaystyle  \begin{array}{rcl}  J(\gamma(A),\gamma(B))=J(\gamma(H),\gamma(K))=\int 1_H P_t 1_K. \end{array}

C’est une inégalité due à Christer Borell (1985), qui utilisait la symétrisation gaussienne, en s’inspirant de Brascamp et Lieb, qui utilisaient le réarrangement (1976). Barthe (1998) a utilisé le transport optimal, l’utilisation de la chaleur dans ce contexte est due à Carlen-Lieb-Loss (2004), Bennett-Carbery-Christ-Tao (2008).

Cette inégalité entra\^{\i}ne l’inégalité isopérimétrique gaussienne.

On peut introduire un potentiel plus convexe que quadratique {V} et la mesure {e^{-V}\,dx}.

5. Le cube discret

Ca a l’air plus simple, mais en réalité, le cube est plus difficile que {{\mathbb R}^n} gaussien. M\^eme {n=1}, {\{-1,1\}} n’est pas si simple.

On a joué avec {X}, {Y} des gaussiennes standard {\rho}-corrélées. L’analogue booléen est un couple {U,V} de variables dont la loi jointe est {(1+\rho xy)\,dx\,dy}.

Quand a-t-on l’inégalité

\displaystyle  \begin{array}{rcl}  \int\int J(f(x),g(y)(1+\rho xy))\,dmu(x)\,d\mu(y)\leq J(\int f\,d\mu,\int g\,d\mu) ? \end{array}

Autrement dit, il s’agit d’une inégalité à 4 points

\displaystyle  \begin{array}{rcl}  \frac{1}{4}J(u,v)+\frac{1}{4}(1+\rho)J(u',v')+\frac{1}{4}(1-\rho)J(u',v)+\frac{1}{4}(1-\rho)J(u,v')\leq J(\frac{u+u'}{2},\frac{v+v'}{2}). \end{array}

Elle est vraie pour {u^\alpha v^\beta} (cela résulte indirectement de l’inégalité de Bonami-Beckner), mais je ne sais pas la démontrer directement. Elle résulte de la {\rho}-concavité. La réciproque est fausse : la fonction du théorème de Borell ne la satisfait pas (prendre {f}, {g} affines et tester sur des dictateurs).

L’analogue du théorème de Borell pour le cube discret consiste à déterminer les ensembles de mesure moitié qui maximisent la stabilité au {\rho}-bruit. Les dictateurs sont systématiquement des contre-exemples, il faut les exclure. C’est le r\^ole de l’influence {I_i(A)=\mu^n\{\}}.

Theorem 1 (Majority is stablest, Mossel-O’Donnell-Oleskiewicz 2010) Soit {M=\{x\in\{-1,1\}^n\,;\,\mathrm{signe }(\sum x_i)=+\}}. Sa stabilité au {\rho}-bruit est {J^B_\rho(\frac{1}{2},\frac{1}{2})} (formule de Sheppard (1899)). Pour tout {\epsilon>0}, il existe {\sigma(\epsilon,\rho)>0} tel que si {\mu^n(A)=\frac{1}{2}} a des influences toutes {\leq\sigma}, alors la stabilité au {\rho}-bruit satisfait

\displaystyle  \begin{array}{rcl}  S_\rho(A)\leq J^B_\rho(\frac{1}{2},\frac{1}{2})+\epsilon. \end{array}

La première preuve passait par le cas gaussien. Une nouvelle démonstration, due à De-Mossel-Neeman (2013), s’appuie sur l’inégalité à 4 points, sauf qu’il y a un reste qu’on peut majorer avec les influences, au moyen de l’hypercontractivité. Cela donne l’impression qu’en changeant de fonction {J}, qui combinerait {J^B} et le {J} de l’hypercontractivité, on pourrait prouver le théorème d’un seul coup.

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Notes of Mario Bonk’s lecture

Dynamics and quasiconformal geometry

1. Motivation : Cannon’s conjecture

{G} Gromov hyperbolic group. When is the ideal boundary a topological 2-sphere ? It is the case when {G} is virtually the fundamental group of a compact hyperbolic 3-manifold.

Version 1: Are there other examples ?

Alternative formulation

Version 2: Show that the ideal boundary {\partial G}, equipped with a visual metric, is quasi-symmetric to the standard 2-sphere.

1.1. Visual metrics

Recall that the visual metric {d(a,b)=e^{-\epsilon (a,b)_p}}, where the Gromov product {(a,b)_p} is (up to a bounded additive error) the distance of base point {p} to the geodesic joing {a} to {b}. Changing the parameter {\epsilon} changes {d} to a snowflake equivalent metric (i.e. a power of it). The word comes from the fact that the von Koch snowflake curve is snowflake equivalent to the real line.

1.2. Quasi-symmetry

Quasi-symmetric means that ratios {\frac{d(x,y)}{d(x,z)}} in one metric are controlled by similar ratios in the other. Equivalently, balls in one metric are pinched between concentric balls in the other, with a bounded ratio of radii. This has to do with quasi-conformality (an infinitesimal version of quasi-symmetry): for Euclidean domains, quasi-symmetry is equivalent to qausi-conformality.

Proposition 1 Under mild assumptions, Gromov hyperbolic metric spaces are quasi-isometric iff their ideal boundaries are quasi-symmetric.

A Gromov hyperbolic group {G} acts on its ideal boundary by quasi-symmetries, in a uniform manner. To make this quantitative, one must inteoduce the notion of quasi-Möbius map (replace ratios {\frac{d(x,y)}{d(x,z)}} with cross-ratios of 4-tuples of points).

Version 2 implies Version 1. Indeed, if {\partial G} is quasi-symmetric to the round sphere, {G} acts by uniformly quasi-Möbius on the round sphere. A result of Sullivan and Tukia implies that the action is by Möbius transformations.

2. The quasi-symmetric uniformization problem

When is a metric space {X} is quasi-symmetric to a standard space {X_0} ?

This is relevant for the

2.1. Kapovich-Kleiner conjecture

Let {G} be a Gromov hyperbolic group whose ideal boundary is homeomorphic to a Sierpinsky carpet (start with a square, cut in 9 pieces, remove central square, iterate in each of the 8 remaining squares). Does {G} arise from a standard situation in hyperbolic geometry ?

This is equivalent to showing that {\partial G} is quassymmetric to a round carpet (remove disjoint geometric circle from a circle, until no interior is left).

Cannon conjecture implies Kapovich-Kleiner conjecture.

This is also relevant to

2.2. Other problems in semi-group dynamics

From a branched covering {f}, one defines a Gromov hyperbolic graph {G_f} (this a rather long story, I willnot give details).

Theorem 2 (Bonk-Meyer, Haissinsky-Pilgrim) Let {f:S^2\rightarrow S^2} be a postcritically finite expanding branched covering. Then {f} is conjugate to a rational map iff {\partial G_f} is quasi-symmetric to the round sphere.

Sometimes it is true, sometimes it is not. The visual metrics arising from branched covering maybe non quasi-symmetric to the standard sphere.

2.3. Example: the snow sphere

Start with the boundary of the cube. Subdivide each face in 9 squares, build a small cube on the middle square, and iterate. This produces a metric which is not a snowflake of the standard sphere (it has rectifiable curves).

Question. Is it quasi-symmetric to the standard sphere ?

At first sight, one would bet that the answer is no. If it were, all squares in the construction should remain uniformly round. Flattening the first stage of the construction to a cube is easy, but doing this at all stages accumulates distorsion. Nevertheless,

Theorem 3 (Meyer) The snow sphere is quasi-symmetric to the round sphere.

3. What is known ?

A lot of positive results for dimensions 0 and 1. No positive results in higher dimensions {n\geq 3}. Semmes writes that all the naive facts one could think of turn out to be wrong. So interesting things happen in dimension 2.

3.1. Low dimensional results

Theorem 4 (Tukia-Väisälä) A metric on the circle is quasi-symmetri to the standard metric iff

  1. It is doubling.
  2. It has bounded turning: every arc {a} with endpoints {x} and {y} has diameter{(a)\leq C\,d(x,y)}.

Similar result for Cantor sets.

3.2. Results in dimension 2

Theorem 5 (Bonk-Kleiner) Let {S} be a metric sphere homeomorphic to the 2-sphere. Assume that

  1. {S} is linearly locally connected (this is a necessary condition).
  2. {S} is Ahlfors 2-regular (this is not at all necessary).

Then {S} is quasi-symmetric to the standard sphere.

Note that visual metrics of hyperbolic groups are Ahlfors-regular (Coornaert).

Theorem 6 (Bonk-Kleiner) Let {G} be Gromov hyperbolic group whose ideal boundary is homeomorphic to the 2-sphere. Assume that the conformal dimension is attained as a minimum. Then {\partial G} is quasi-symmetric to the standard sphere.

Recall that the conformal dimension is the infimal dimension of Alhfors regular metric spaces quasi-symmetric to {\partial G}.

We strongly use 2 dimensions, but an intermediate step applies in all dimensions: conformal dimension attained implies metric is Löwner. Note that there are examples (due to Bourdon and Pajot) of groups whose conformal dimension is not attained. These examples have boundaries which are not 2-spheres.

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Notes of Zoltan Balogh’s lecture nr 2

Horizontal convexity in the Heisenberg group

Joint with Andrea Calogero and Alexandru Kristály.

1. Alexandrov’s theorem in Euclidean space

I think of a convex function as a function which stays above its supporting affine function at every point. The set of slopes (covectors) of these affine functions at {x_0} is called the subdifferential {\partial u(x_0)} of {u} at {x_0}. The union of all these sets over the domain is denoted by {\nabla u(\Omega)}. Its measure {\mathcal{L}^n(\nabla u(Omega))} is sometimes called the Monge-Ampère measure of {u}.

Theorem 1 (Alexandrov) There is a dimension dependant constant {C} with the following effect. Let {\Omega} be an open, bounded convex domain in {{\mathbb R}^n}. Let {u} be a convex function on the closure of {\Omega}, which vanishes on the boundary. Then, for all {x_0\in\Omega},

\displaystyle  \begin{array}{rcl}  |u(x_0)|^n\leq C\,d(x_0,\partial \Omega).\mathrm{diameter}(\Omega)^{n-1}.\mathcal{L}^n(\nabla u(Omega)). \end{array}

This is used in PDE (Caffarelli,…).

2. Convexity in Heisenberg group

Since left translations ar affine is exponential coordinates, Heisenberg group carries an affine structure. Therefore convex domains will simply be Euclidean convex.

Definition 2 (Several competing groups) Say a function on a convex domain {\Omega} of Heisenberg group is {H}-convex if its restriction to every horizontal line of {\Omega} is convex.

The subdifferential of {u} at {x_0} is a subset of {{\mathbb R}^{2n}}.

Note that there are H-convex functions in {\mathbb{H}^n} which are very irregular (e.g. Weierstrass) in the vertical direction.

2.1. Results

We define a horizontal slicing diameter : this is the maximal diameter of the intersection of {\Omega} with horizontal planes {H_x}, {x\in\mathbb{H}^n}. We also define a horizontal slicing Monge-Ampère measure

\displaystyle  \begin{array}{rcl}  \mathcal{L}_{HS}^{2n}(\nabla_H u(\Omega))=\sup_{x\in\Omega}\mathcal{L}^{2n}(\nabla_H u (H_x \cap\Omega)). \end{array}

Theorem 3 There is a dimension dependant constant {C} with the following effect. Let {\Omega} be an open, bounded convex domain in {\mathbb{H}^n}. Let {u} be a convex function on the closure of {\Omega}, which vanishes on the boundary. Then, for all {x_0\in\Omega},

\displaystyle  \begin{array}{rcl}  |u(x_0)|^{2n}\leq C\,d(x_0,\partial \Omega).\mathrm{diameter}_{HS}(\Omega)^{2n-1}.\mathcal{L}_{HS}^{2n}(\nabla u(\Omega)). \end{array}

This improves earlier results by Garofalo et al. where the distance to the boundary appeared with a negative power.

3. Proof

3.1. Back to the Euclidean case

Lemma 4 (Comparison principle) Let {u}, {v} be continuous functions on the closure of {\Omega}. Assume that {u\leq v}. Then

\displaystyle  \begin{array}{rcl}  \nabla v(Omega)\subset \nabla u(\Omega). \end{array}

Indeed, any supporting hyperplane of the graph of {u}, when raised, will touch the graph of {v}.

Alexandrov compares the graph of {u} with the cone on {\partial \Omega} with vertex at {(x_0,u(x_0))}. Its subdifferential is concentrated at the vertex. Let {x} be the nearest point in the boundary. In the subdifferential {\partial v(x_0)}, there is a covector {p_1} of size {\sim|u(x_0)|/d(x,x_0)}. All othe covectors {p} in {\partial v(x_0)} satisfy

\displaystyle  \begin{array}{rcl}  |p|\geq \frac{|u(x_0)|}{\mathrm{diameter}(\Omega)}. \end{array}

3.2. Failure of comparison principle in Heisenberg group

There exists functions {u,v} on a cyclinder {\Omega}, which are equal n the boundary and {u\leq v}, but {\nabla_H v(\Omega)\not\subset \nabla_H u(\Omega)}.

Indeed, set {v(x,y,t)=t}. Check that {\partial _H v(x,y,t)=(2y,-2x)}, so that {\nabla_H v(\Omega)=B(0,2)} contains the origin. Modify {v} in an annulus,

\displaystyle  \begin{array}{rcl}  u(x,y,t)=t-(1-t^2)g(x,y) \end{array}

where {g} has support in an annulus. Assume that {0\in \nabla_H u(\Omega)}. Then {0\in \partial_H u(q)}, and {u} achieves its minimum on {H_q} at point {q}. One can achieve that this never happens.

3.3. Comparison for convex functions

What saves us is that comparison holds for convex functions.

Theorem 5 Let {\Omega} be a convex domain in {\mathbb{H}^n}, let {u,v} be convex functions on {\Omega} that are equal on the boundary. Assume that for some {x_0\in\Omega}, there exists {p\in\partial_H v(x_0)} such that, for al {x\in H_{x_0}\cap\Omega} different from {x},

\displaystyle  \begin{array}{rcl}  v(x) > v(x_0)+p.(\pi(x)-\pi(x_0)). \end{array}

Then {p\in \partial_H u(x_0)}.

The proof uses degree theory for set valued maps. For simplicity, let us assume that {u} is smooth, and {x_0=0}. Let {U=H_{x_0}\cap \Omega} projected to {{\mathbb R}^{2n}}. We view {\partial_H u} as a mapping of {U} to {{\mathbb R}^{2n}}. To show that {p} belongs to its image, it suffices to show that the degree of {\partial_H u} on {U} at {p} is non zero. We check that this is the case when {\partial_H u} is replaced with {\partial_H v}. Then a linear homotopy allows to conclude. Indeed, assume by contradiction that the homotopy hits {p} along {\partial U}, i.e. there exists a point {x\in\partial\Omega\cap H_{x_0}} and {t\in[0,1]} such that

\displaystyle  \begin{array}{rcl}  t\partial_H u(x)+(1-t)\partial_H v(x)=\partial_H v(x_0). \end{array}

Along the horizontal line from {x_0} to {x},

\displaystyle  \begin{array}{rcl}  u(x_0) \geq u(x)+\partial_H u(x).(\pi(x_0)-\pi(x)),\quad v(x_0) \geq v(x)+\partial_H v(x).(\pi(x_0)-\pi(x)). \end{array}

Take the convex combination of these two inequations, get and inequality that contradicts the assumption {v(x) > v(x_0)+p.(\pi(x)-\pi(x_0))}.

Computation of the index for {v}.

3.4. End of the proof

One gets

\displaystyle  \begin{array}{rcl}  |u(x_0)|^{2n}leq C\,d(x_0,H_{x_0}\cap\partial \Omega).\mathrm{diameter}_{HS}(\Omega)^{2n-1}.\mathcal{L}_{HS}^{2n}(\nabla u(Omega)). \end{array}

There remains to replace {d(x_0,H_{x_0}\cap\partial \Omega)} with {d(x_0,\partial \Omega)}. This relies of an Harnack inequality, which allows to replace {x_0} with a nearby point where the horizontal plane is tilted and hits the boundary at a distance comparable to the distance of {x_0} to the boundary.

Next sessions : April 4th and May 14th.

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Notes of Francois Vigneron’s lecture

Multifractal analysis on the Heisenberg group

Joint with Stéphane Seuret.

1. Motivation

1.1. Sources of multifractal analysis

Multifractal analysis is a toolbox for data analysis (textures, financial or experimental data, diophantine approimation,… see the program of our seminar at UPEC). It introduces classification parameters based on absence of regularity. Why the Heisenberg group ? People from image analysis ask about non isotropic textures.

1.2. Historic examples

Weierstrass’ example of a function {W_h} which is {C^h}-Hölder but nowhere differentiable (for every {0<h<1}.

Riemann function which is differentiable at infinitely many rational points (but not all of them) and nowhere else,

\displaystyle  \begin{array}{rcl}  \sum_n n^{-2}\sin(2\pi n^2 x). \end{array}

Its multifractal spectrum was not computed before 1996 (S. Jaffard).

Both examples are self similar. A picture of Weierstrass’ function suggests that irregularity is spread all over. A picture of Riemann’s function looks very different : spikes, points with different left and right derivatives, differentiability point, all over.

Definition 1 The pointwise regularity exponent {h_f(x_0)} of {f} at {x_0} is the largest {\alpha>0} such that {f} is, up to a polynomial, {O(|x-x_0|)^\alpha)}.

The multifractal spectrum is the function

\displaystyle  \begin{array}{rcl}  d_f(h)=\mathrm{dim}_{\mathrm{Hausdorff}}(\{x\,;\, h_f (x)=h\}). \end{array}

Example 1 Weierstrass function {W_h} is monofractal : every point has pointwise regularity exponent {h}.

The multifractal spectrum of Riemann’s function is made of a segment joining {h=1/2} and {h=3/2} plus a point at {h=3/2}.

2. Heisenberg group

The strong anisotropy turns out not the make a big change for the specific questions we solve the existing techniques adapt rather easily. This is why we can state theorems

2.1. Wavelets

One can construct smooth, exponentially decaying functions {\psi_{j,k}^{\epsilon}} concentrated at {2^{-j}\circ k}, where {k\in {\mathbb Z}^3} (which is a subgroup), with vanishing moments, which form a basis of {L^2}.

2.2. Results

Theorem 2 (Global Hölder regularity) Let {s=k+\sigma}. A function belongs to {C^s} iff its {k}-th horizontal derivatives are {C^\sigma}. Also iff its wavelet coefficients

Theorem 3 (Pointwise Hölder regularity) Let {s=k+\sigma}. If a function {f} is {C^s} at {x_0}, then

\displaystyle  \begin{array}{rcl}  2^{js}|d_{j,k}^\epsilon(f)|\leq C(1+2^j d(x_{j,k},x_0))^s . \end{array}

Conversely, if this holds, then {f} is {C^t} at {x_0} for all {t<s}.

2.3. Generic spectrum in H\” older and Besov classes

Theorem 4 Monofractal functions (at {s}) form a dense {G_\delta} subset of {C^s}.

Definition 5 {f\in B_{p,q}^s} if

\displaystyle  \begin{array}{rcl}  \|2^{j(s-Q/p)}|d_{j,k}^\epsilon(f)|\|_{\ell^p(k)}\in \ell^{q}(j) . \end{array}

Note that {B_{p,q}^s \subset C^{s-Q/p}} for {s>Q/p} and {s\notin Q/p +{\mathbb N}}. Here, {Q=4}.

Theorem 6 For a dense {G_\delta} subset of {B_{p,q}^s}, the spectrum is a segment between {(s-Q/p,0)} and {(s,Q)}. For all functions of {B_{p,q}^s}, the spectrum is below this segment.

Indeed, for the standard example of a Besov function (expressed in wavelet expansion), the pointwise Hölder exponent at {x_0} is related to the dyadic approximation rate of {x_0}. Dimensions of isoapproximable sets can be computed. It is a special case of a very general result by Beresnevich, Dickinson and Velani (2006).

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