Notes of Anton Thalmaier’s lecture nr 1

The geometry of subelliptic diffusions

1. Stochastic flows

Let {A} be a vectorfield with flow {(\phi_t)}. For compactly supported functions {f},

\displaystyle  \begin{array}{rcl}  f\circ\phi_t(x)-f(x)-\int_{0}^{t}A(f(\phi_s(x)))\,ds=0. \end{array}

Can one attach a flow to a second order operator ? E.g. to

\displaystyle  \begin{array}{rcl}  L=A_0+\sum A_i^2. \end{array}

Basic example is the Euclidean Laplacian. Answer is yes, but flow lines now depend on a random parameter {\omega}, {\phi_t(x,\omega)}. Also, they are no more differentiable as function of {t}. In other words, the flow becomes a stochastic process {X_t(x)=\phi_t(x,\omega)}.

1.1. Formal definition

The data are a filtered probability space, i.e. a probability space {(\Omega,\mathcal{F},P)} equipped with an increasing {\sigma}-algebra {\mathcal{F}_t\subset\mathcal{F}}. Think of {\mathcal{F}_t} as representing the events having occurred up to time {t}.

An adapted continuous process is a family of random variables {X_t(x)}, {\mathcal{F}_t}-measurable, with a.e. continuous trajectories. It is a flow process of {L}, (or an {L}-diffusion) with starting point at {x} if {X_0(x)=x} and, for compactly supported functions {f},

\displaystyle  \begin{array}{rcl}  N_t^f(x):=f(X_t(x))-f(x)-\int_{0}^{t}(Lf)(X_s(x)))\,ds \end{array}

is a martingale, i.e. for every {s\leq t}, the conditional expectation

\displaystyle  \begin{array}{rcl}  \mathop{\mathbb E}(N_t^f(x)-N_s^f(x)|\mathcal{F}_s)=0. \end{array}

In other words, {N_t^f} has no specific trend, it is only fluctuations. One can also say that {N_s^f(x)} is the best possible prediction of {N_t^f(x)} one can do with the knowledge one has at time {s}.

The operator {f\mapsto P_t(f)=\mathop{\mathbb E}(f(X_t))} is a semi-group. Since {\mathop{\mathbb E}(N_t^f(x))=\mathop{\mathbb E}(N_0^f(x))=0}, the flow law implies that

\displaystyle  \begin{array}{rcl}  \frac{d}{dt}P_t (f)=P_t(Lf). \end{array}

In particular, the governing differential operator {L} is recovered by

\displaystyle  \begin{array}{rcl}  \frac{d}{dt}P_t (f)_{|t=0}=Lf. \end{array}

Remark. The life time of the process may be finite. It is a stopping time {\zeta(x)}. We assume that {\zeta(x)=\infty} implies that {X_t(x)} tends to infinity. Then, for functions {f} which need not be compactly supported, {N_t^f(x)} is a local martingale, i.e. it becomes a martingale when stopped at {\zeta(x)}.

1.2. Basic example

Let {L=\frac{1}{2}\Delta} be half the Laplacian. The corresponding flow is the Brwnian motion {B_t}. It\^o’s formula states that

\displaystyle  \begin{array}{rcl}  f(X_t)-f(X_0)+\int_{0}^{t}\langle\nabla f(X_s),dX_s\rangle +\frac{1}{2}\int_{0}^{t}\Delta f(X_s)\,ds. \end{array}

The first term is a stochastic integral. It is a martingale.

What is this good for ? Here are a few applications.

1.3. The Dirichlet problem

The problem: let {D} be an open set, {\phi} a continuous function on the boundary {\partial D}. Find a continuous extension {u} to {\bar{D}} which is {L}-harmonic, i.e. {Lu=0}.

Assume that there exists an {L}-diffusion {X_t} with a.s. finite life time. Assume {u} is a solution to the Dirichlet problem. Exhaust {D} with compact sets {D_n}. Let {u_n} be a compactly supported function that coincides with {u} on {D_n}. Let {\tau_n(x)} denote the exit time from {D_n}. Then, for {x\in D_n},

\displaystyle  \begin{array}{rcl}  N_t(x):=u_n(X_t(x))-u_n(x)-\int_{0}^{t}(Lu_n)(X_s(x)))\,ds \end{array}


\displaystyle  \begin{array}{rcl}  0=\mathop{\mathbb E}(N_{t\wedge \tau_n(x)}(x))=\mathop{\mathbb E}(u_n(X_{t\wedge \tau_n(x)}(x)))-\mathop{\mathbb E}(u_n(x))=\mathop{\mathbb E}(u(X_{t\wedge \tau_n(x)}(x)))-u(x). \end{array}


\displaystyle  \begin{array}{rcl}  u(x)=\mathop{\mathbb E}(u(X_{t\wedge\tau_n(x)}(x))). \end{array}

and taking a limits, first {n} to infinity, then {t} to infinity,

\displaystyle  \begin{array}{rcl}  u(x)=\mathop{\mathbb E}(u(X_{\tau(x)}(x)))=\mathop{\mathbb E}(\phi(X_{\tau(x)}(x))), \end{array}

since, by assumption, {\tau(x)<\infty} a.s., i.e. {\phi(X_{\tau(x)}(x))\in\partial D} a.s.

In other words, we get uniqueness of the classical solution of Dirichlet’s problem under the single condition {\tau(x)<\infty} a.s. It also leads to an efficient numerical method for computing the solution, the Monte-Carlo method.

Conversely, define a function {u} by {u(x)=\mathop{\mathbb E}(\phi(X_{\tau(x)}(x)))}. It is true that {u} is {C^2} and {Lu=0}. In order to prove that {u} extends continuously to {\partial D}, one needs that {\tau(x)} tends to 0 in probability as {x} tends to a point of {\partial D}.

Example 1 {L=\partial_{\theta}^2} on annulus has no uniqueness.

Indeed, any radial function vanishing on the boundary is {L}-harmonic. In fact, {X_t} is 1-dimensional Brownian motion on each circle. It never exists, so {\tau\equiv\infty}.

Example 2 {L=\partial_{x}^2} on a symmetric bean shaped planar domain {D} has no existence.

With a boundary data symmetric on the convex part, any solution takes values determined by the convex boundary. Indeed, {\tau(x)} does not tend to 0 as {x} tends to the point where the concave boundary touches the {x} axis.

1.4. Heat equation

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Notes of Nicola Garofalo’s lecture nr 2

1. Stratified nilpotent groups

aka Carnot groups.

1.1. Examples: Heisenberg groups

It was known to physicists under the name Weyl’s group. It was re-christened Heisenberg group by Elias Stein and his school of harmonic analysis.

{\mathbb{H}^n} is a multiplication on {\mathbb{C}^n\times{\mathbb R}}. The Lie algebra {\mathfrak{h}^n=\mathbb{C}^n\oplus{\mathbb R}:=V_1\oplus V_2} with {[V_1,V_1]=V_2}, all other brackets vanishing. The bracket on {V_1} is the symplectic form {\Im m(z\cdot\bar{z'})}. Spelling {z=x+iy},

\displaystyle  \begin{array}{rcl}  [x+iy,x'+iy']=\frac{1}{2}\sum_{j}x_jy'_j-x'_jy_j. \end{array}

We assign a formal degree 1 to {V_1} and 2 to {V_2} and define the nonisotropic Lie algebra dilation

\displaystyle  \begin{array}{rcl}  \Delta_\lambda(z,t)=(\lambda z,\lambda^2 t). \end{array}

The nonisotropic group dilation {\delta_\lambda} is given by the same formula. Its Jacobian equals {\lambda^{2n+2}}. The exponent {Q=2n+2} is called homogeneous dimension, it will play the role of dimension in Euclidean analysis.

1.2. Carnot groups

A stratified nilpotent Lie algebra of step {r} takes the form {\mathfrak{g}=V_1\oplus\cdots\oplus V_r} with {[V_1,V_j]=V_{j+1}}, {j=1,\ldots,r-1}, and {[V_1,V_r]=0}.

A Lie group {G} is a Carnot group of step {r} if the corresponding Lie algebra {\mathfrak{g}} is stratified nilpotent of step {r}.

Define {\Delta_\lambda(\xi_1+\cdots+\xi_r)=\lambda\xi_1+\cdots+\lambda^r\xi_r}. These are Lie algebra automorphisms. Since {G} is nilpotent, the group exponential map {\exp:\mathfrak{g}\rightarrow G} is an analytic diffeomorphism. So {\delta_\lambda=\exp\Delta_\lambda\exp^{-1}} are group automorphisms, the nonisotropic group dilations.

Fix an inner product on {\mathfrak{g}} that makes {V_j}‘s orthogonal. The nonisotropic gauge on {\mathfrak{g}} is

\displaystyle  \begin{array}{rcl}  |\xi|=(\sum\|\xi_s\|^{2r!/s})^{1/2r!}. \end{array}

It is homogeneous of degree one under dilations {\Delta_\lambda}. This is not a norm in the usual sense. Never mind. Using {\exp}, we get a nonisotropic gauge on {G}, which is homogeneous under {\delta_\lambda}.

The Baker-Campbell-Hausdorff formula expresses the multiplication rule of {G} in exponential coordinates,

\displaystyle  \begin{array}{rcl}  \exp^{-1}(\exp(\xi)\exp(\eta))=\xi+\eta+\frac{1}{2}[\xi,\eta]+\frac{1}{12}([\xi,[\xi,\eta]]-[\eta,[\xi,\eta]])+\cdots \end{array}

The full series can be found in books. For nilpotent groups, it is a finite sum. Exercise: compute the multiplication for {\mathbb{H}^1}.

1.3. The sub-Laplacian

Let {e_1,\ldots,e_m} be an orthonormal basis of {V_1}. They represent left-invariant vectorfields on {G}: {X_i(g)=(L_g)_*(e_j)} where {L_g} is left translation by {g}, {L_g(g')=gg'}. Define

\displaystyle  \begin{array}{rcl}  \Delta_{H}=\sum X_j^2. \end{array}

It depends on the choice of orthonormal basis. Thanks to Hörmander’s theorem, {\Delta_H} is hypoelliptic. It is left-invariant and homogeneous of degree 2 under dilations.

1.4. The {J}-map

For step 2 groups, the following map is useful. Assume that the chosen inner product on {\mathfrak{g}} makes {V_1} and {V_2} orthogonal. {J:V_2\rightarrow End(V_1)} is defined by

\displaystyle  \begin{array}{rcl}  \langle J(t)z,z'\rangle=\langle[z,z'],t\rangle. \end{array}

For convenience, fix an orthonormal basis {\epsilon_1,\ldots,\epsilon_k} of {V_2}.

Proposition 1 Let {G} be a step 2 Carnot group. Then

\displaystyle  \begin{array}{rcl}  X_j&=&\partial_{z_j}+\frac{1}{2}\sum\langle J(\epsilon_\ell),e_j\rangle\partial_{t_\ell},\\ \Delta_H&=&\Delta_z+\frac{1}{4}\sum_{\ell,\ell'}\langle J(\epsilon_\ell)z,J(\epsilon_{\ell'}z\rangle\cdot\partial^{2}_{\ell,\ell'}+\sum_{\ell}\partial_{t_\ell}\sum_i\langle J(\epsilon_\ell)z,e_i\rangle \partial_{z_i}. \end{array}

1.5. Groups of Métivier type

These are the step 2 Carnot groups for which {J} is nondegenerate,

\displaystyle  \begin{array}{rcl}  |J(t)z|\geq\mathrm{const.}\,|z||t|. \end{array}

1.6. Groups of Heisenberg type

These are the step 2 Carnot groups for which {J} is isometric,

\displaystyle  \begin{array}{rcl}  \langle J(t)z,J(t)z'\rangle = |t|^2\langle z,z'\rangle. \end{array}

They were introduced by Aroldo Kaplan in research concerning hypoellipticity. Although it is a much smaller class that Métivier’s, it turns out to be quite rich. It contains quaternionic Heisenberg groups and many variants. It played a historical role in progress in understanding hypoellipticity.

For such groups, the formula for {\Delta_H} simplifies,

\displaystyle  \begin{array}{rcl}  \Delta_H=\Delta_z+\frac{|z|^2}{4}\Delta_t+\sum_{\ell}\sum_i\langle J(\epsilon_\ell)z,e_i\rangle \partial_{z_i}\partial_{t_\ell}. \end{array}

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Notes of Ludovic Rifford’s lecture nr 2

1. Sub-Riemannian geodesics

1.1. Sub-Riemannian structures

A sub-Riemannian structure is the data of a manifold {M}, a smooth distribution of subspaces {\Delta(x)\subset T_xM} of constant rank {m}, and a smoothly varying scalar product on {\Delta(x)}.

Example: restriction of a Riemannian metric to {\Delta}.

Locally, {\Delta(x)=\mathrm{span}(X_1,\ldots,X_m)}, but it may be impossible to get such a frame globally.

Example: left-invariant distribution

A horizontal path is a {W^{1,2}} map from an interval to {M} whose derivative {\dot{\gamma}(t)\in\Delta(\gamma(t))} a.e. Thus locally, horizontal paths coincide with trajectories of a control system wih {L^2} controls.

1.2. Sub-Riemannian distances

Define the sub-Riemannian distance {d_{SR}} by minimizing the length of horizontal paths joining points. It defines the usual topology of {M}.

The energy of a horizontal path is the squared {L^2}-norm of its derivative.

A minimizing geodesic is a horizontal path which minimizes energy among all horizontal paths joining its endpoints. Then speed is constant and length is minimized.

Theorem 1 If {d_{SR}} is complete, closed balls are compact and every two points are joined by a minimizing geodesic.

Example: restricting a complete Riemannian metric to {\Delta} produces a complete sub-Riemannian distance.

1.3. Calculus of variations

Locally, one can use an orthonormal frame. Let {E=E_x} denote the corresponding endpoint map starting at {x}. Minimizing geodesics from {x} to {y} are trajectories of controls {u} which minimize the squared {L^2} norm functional {C} under the constraint {E(u)=y}. The Lagrange Multiplier Theorem implies that there are {\lambda_0\in\{0,1\}} and {p\in T_y^*M}, {(\lambda_0,p)\not=(0,0)}, such that

\displaystyle  \begin{array}{rcl}  p\cdot D_u E=\lambda_0 D_u C. \end{array}

There are two cases, {\lambda_0=0} or {\lambda_0=1}.

If {\lambda_0=0}, {u} is singular. This case occurs but is not that frequent. Tomorrow, I will study in detail a number of examples, where singular controls occur but never along minimizaing geodesics, for instance. Note that it is unclear wether such a singular minimizing control needs to be smooth.

1.4. The Hamiltonian geodesic equation

From now on, {\lambda_0=1}. Define the Hamiltonian

\displaystyle  \begin{array}{rcl}  H(x,p)=\frac{1}{2}\sum(p\cdot X^i(x))^2 . \end{array}

Proposition 2 There is a smooth map {p:[0,1]\rightarrow T^*M} with {p(1)=\frac{p}{2}}, called the normal extremal lift of the geodesic, such that {p} is a trajectory of the Hamiltonian vectorfield associated to {H}, i.e.

\displaystyle  \begin{array}{rcl}  \dot{\gamma}&=&\frac{\partial H}{\partial p},\\ \dot{p}&=&-\frac{\partial H}{\partial x}. \end{array}

In particular, the control is smooth.

The proof uses the variational equation and {DE}.

In summary, minimizing geodesics

  • either are singular,
  • or admit a normal extremal lift,
  • or both.

1.5. Example: Heisenberg group

The control system is

\displaystyle  \begin{array}{rcl}  \dot{x}&=&u_1\\ \dot{y}&=&u_2\\ \dot{z}&=&\frac{1}{2}(u_2 x-u_1 y). \end{array}

Integrating gives

\displaystyle  \begin{array}{rcl}  z(1)-z(0)=\int_{\alpha}\frac{1}{2}(xdy-ydx)=\int_{D}dx\wedge dy +\int_{c}\frac{1}{2}(xdy-ydx), \end{array}

where {\alpha} is th projection of the trajectory in the plane, {c} a line segment closing it and {D} the planar domain they surround. Minimizing length amounts to solving an isoperimetric problem in the plane. The solution is known to be an arc of circle.

This can be seen by solving the Hamiltonian equations. They imply that {p_z} is constant,

\displaystyle  \begin{array}{rcl}  \ddot{x}=-p_z\dot{y},\quad \ddot{y}=p_z\dot{x}. \end{array}

If {p_z=0}, one finds lines. If {p_z\not=0}, one finds circles.

1.6. Example: The Martinet distribution

It is the kernel of {dz-x^2 dy}, spanned by orhtonormal basis {X_1=\partial_x}, {X_2=(1+x)\partial_y+x^2\partial_z}. It turns out that {\gamma(t)=(0,t,0)} is a singular minimizing geodesic which has no normal extremal lift.

1.7. The sub-Riemannian exponential mapping

It is defined on an open subset {\mathcal{E}_x} of {T_x^* M}, it maps a covector at {x} to the footpoint of its image by the time 1 flow of the Hamiltonian {H}.

Proposition 3 If {d_{SR}} is complete, then for all {x}, {\mathcal{E}_x=T_x^* M}.

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Notes of Nicola Garofalo’s lecture nr 1

Hypoelliptic operators and analysis on Carnot-Carathéodory spaces

1. Hypoelliptic operators

1.1. Motivation: semi-flexible polymers

In 1995, when studying Euler’s elastica, introduced the following differential operator

\displaystyle M=\partial_\theta^2+\cos\theta \partial_x +\sin\theta \partial_y -\partial_t.

It turns out to play a role in models of semi-flexible polymers.

Write {X_1=\partial_\theta} and {X_0=\cos\theta \partial_x +\sin\theta \partial_y -\partial_t}. Then {X_2:=[X_1,X_0]=-\sin\theta \partial_x +\cos\theta \partial_y}, {X_3:=[X_1,X_2]=-\cos\theta \partial_x -\sin\theta \partial_y}. Thus {X_1} and {X_0} are bracket generating.

1.2. Hypoellipticity

A differential operator {P} with smooth coefficients is hypoelliptic if {Pu=f} with {f} smooth and {u} a distribution implies that {u} is smooth.

The main example is the Laplacian {\Delta} (sometimes known as Weyl’s Lemma, due to Cacciopoli in 1938, generalized to variable coefficients by Cimino in 1940).

The next example is the heat operator {\Delta-\frac{\partial}{\partial t}}. Note that it is not {C^\omega}-hypoelliptic. On the other hand, the wave equation {\Delta-\frac{\partial^2}{\partial t^2}} is not hypoelliptic.

Theorem 1 (Hörmander 1967) If {X_0,X_1,\ldots,X_m} are smooth bracket generating vectorfields and {c} is a smooth function, then

\displaystyle  \begin{array}{rcl}  L=\sum X_i^2 +X_0+c \end{array}

is hypoelliptic.

The bracket generating condition is nearly necessary, as shown by Hörmander in his PhD in 1954 (under Gårding).

Note that higher order differential operators are not hypoelliptic.

1.3. Back to Mumford’s operator

Write {z=x+iy} and introduce the group law

\displaystyle  \begin{array}{rcl}  (\theta,z,t)(\theta',z',t')=(\theta+\theta),z+e^{i\theta}z',t+t'). \end{array}

This is the Lie group {RT\times{\mathbb R}}, where {RT} denotes the planar roto-translation group. Then Mumford’s operator is left-invariant. In fact, each {X_i} is left-invariant. Note that {RT} is not nilpotent, this group does not have dilations.

1.4. Kolmogorov’s operator

In 1934, in his approach to the kinetic theory of gases, Kolmogorov introduces the equation

\displaystyle  \begin{array}{rcl}  Ku=\partial_x^2 +x\partial_y -\partial_t=X_1^2+X_0 \end{array}

where {X_1=\partial_x} and {X_0=x\partial_y -\partial_t}, {X_2:=[X_1,X_0]=\partial_y} generate {{\mathbb R}^3}. So Kolmogorov’s operator is hypoelliptic (this conclusion is one of Hörmander’s main motivations). In fact, Kolmogorov’s had computed an explicit fundamental solution for {K}, which is smooth outside the diagonal, this implies hypoellipticity.

1.5. Stein’s program

Probabilists, starting from Mark Kac, realized very early that {M} and {K} are related in the same way as a manifold is connected to its tangent space.

In his ICM 1970 address, Stein launched a program of developping noncommutative harmonic analysis by approximating operators by their second order Taylor expansions.

Expanding {\cos} and {\sin} at second order, we approximate {M} with {L=\partial_\theta^2+\frac{\theta^2}{2}\partial_x+\theta\partial_y-\partial_t}, which is again hypoelliptic, by Hörmander’s theorem. Switch notation to make {L=X_1^2+X_0} where {X_1=\partial_x}, {X_0=\frac{x^2}{2}\partial_y+x\partial_z-\partial_t}. Define

\displaystyle  \begin{array}{rcl}  \delta_\lambda(x,y,z,t)=(\lambda x,\lambda^4 y,\lambda^3 z,\lambda^2 t). \end{array}

Then {(\delta_\lambda)_*L=\lambda^2 L}.

1.6. Exponential map and group law for {L=X_1^2+X_0}

A theorem of Lanconelli states that there is a Lie group underlying every real analytic differential operator admitting dilations, under some bracket generating condition. We perform the calculation for {L}.

\displaystyle  \begin{array}{rcl}  Exp(uY)(g)=\begin{pmatrix} x+u_1\\y+\frac{u_2}{2}(x^2+\frac{u_1^2}{3}+u_1 x)+u_3 x +\frac{u_1 u_3}{2}+u_4\\z+u_3+u_2 x+\frac{u_1 u_2}{2}\\t-u_2 \end{pmatrix} \end{array}

From the exponential map, one extracts the group law in {{\mathbb R}^4},

\displaystyle  \begin{array}{rcl}  \begin{pmatrix} x\\y\\z\\t \end{pmatrix}\begin{pmatrix} x'\\y'\\z'\\t' \end{pmatrix}=\begin{pmatrix} x+x'\\y+y'+xz'-\frac{t'x^2}{2}\\z+z'-t'x\\t+t' \end{pmatrix}. \end{array}

The {X_i}‘s become left-invariant.

Kolmogorov’s operator differs from {L}, the associated Lie group is different. A third operator, similar to {K} and {L}, has been studied recently by Citti, Menozzi and Polidoro.

2. Stratified nilpotent Lie groups

Also known as Carnot groups.

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Notes of Ludovic Rifford lecture nr 1

Geometric control and sub-Riemannian geodesics

Contents of the course

  1. The Chow-Rashevsky Theorem
  2. Sub-Riemannian geodesics
  3. A closer look at singular curves


Here is recommended reading related to the contents of this course.

  • Bellaïche: The tangent space in sub-Riemanian geometry
  • Montgomery: A tour of subriemannian geometries
  • Agrachev-Barilari-Boscain: Introduction to Riemannian and sub-Riemannian geometry, to appear soon
  • Jean: Control of nonholonomic systems: from sub-Riemannian geometry to motion planning
  • Rifford: Sub-Riemannian Geometry and optimal transport

1. Control systems

Example: reversed pendulum on a cart.

In general, a control system is {\dot{x}=f(x,u)}, {x} is the state of the system, {u} is the control, both are functions of time with values in vectorspaces.

Example: given vectorfields {X_1,\ldots,X_m}, {f(x,u)=\sum u_i X_i(x)}.

1.1. Controllability

Controllability: which states can one reach from a given state {x_1} ?

Theorem 1 (Chow 1939-Rashevski 1938) Data: vectorfields {X_1,\ldots,X_m} on a neighborhood. Assume that the Lie algebra generated by these generates the whole tangent space at point {x_1}. Then every point in a smaller neighborhood of {x_1} can be reached from {x_1}.

The conclusion is called local controllability. Note that on a connected manifold, local controllability implies global controllability.

1.2. Geometric interpretation

The theorem helps understanding the meaning of Lie brackets: {[X,Y]} is a direction in which you can move using controls in directions {X} and {Y} only. Moving in that direction requires patience: moving aling {X}, {Y}, {-X} and finally {-Y} produces a quadrilateral which nearly closes up, but not quite. It is the small defect which is in the direction {[X,Y]}, so it takes lots of time to go there.

1.3. Number of steps

The assumption in the theorem bears many different names (one in each of the above monographs), let us call it bracket generating. Let {\mathcal{F}=\{X_1,\ldots,X_m\}} be a family of vectorfields. Denote by {Lie^{1}(\mathcal{F})=span(\mathcal{F})} and then, recursively,

\displaystyle  \begin{array}{rcl}  Lie^{k+1}=Lie^{k}(\mathcal{F})\cup\{[X,Y]\,;\,Y\in Lie(\mathcal{F}),\,Y\in Lie^{k}(\mathcal{F})\}. \end{array}

Bracket generating means that for some {r\in{\mathbb N}}, {Lie^{k}(\mathcal{F})} evaluated at {x_1} equals the tangent space. The smallest {r} has geometric sgnificance.

1.4. Ways of proving the theorem

In {{\mathbb R}^3}, here is a proof of the theorem. In the obvious commutator of two flows, introduce a small third parameter. Show that one gets a local diffeo.

This works in all dimensions, see Jean’s monograph. It even gives a stronger conclusion known as the ball-box theorem.

2. Proof of the Chow-Rashevski Theorem

Nevertheless, I will follow Bellaïche’s proof.

2.1. Singular controls

Introduce the end-point mapping {E} from the space of controls to the space of states. Its differential is given by the variational equation, a linear ODE. We say that a control {u} is regular if {E} is a submersion at {u}, singular otherwise. In fact, it is merely a property of the trajectory as a curve. Reparametrization (and in particular, time reversal) keeps a trajectory regular or singular.

The rank of a singular control {u} is the rank of {E} at {u}.

The Chow-Rashevski Theorem will follow from the following

Proposition 2 For a bracket generating family of vectorfields, the end-point map is open.

2.2. Density of regularity

Most controls in {L^2} are regular. In fact, every ball of {L^2} contains a regular control. Proof by contradiction. Let {u} maximize {rank(DE)} in a small ball of {L^2}, let {d<n} be its rank. There exists controls {v_i} such that {\lambda\mapsto E(u+\sum\lambda_{i=1}^d v_i)} is an immersion at 0, its image is a submanifold {N}. Then {E} maps into {N}. In particular, the initial vectorfields {X_i} are tangent to {N}, contradiction.

More generally, the set of regular controls is dense in {L^2} (concatenate). A theorem of Sontag asserts that is dense in {C^\infty}.

2.3. The return method

The idea is that if {u} is regular, {u} concatenated with its time reversal is regular.

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Notes of Camille Horbez’ lecture

Horoboundary of Outer space, and growth under random automorphisms

1. Random growth

Question. Pick an element {g} of free group {F}. Apply a sequence of random elements of {Aut(F)}. How fast does the length grow after cyclic reduction ?

Theorem 1 Let {g\in F}. Let {\mu} be a probability measure on {Out(F)} whose support is finite and generates {Out(F)}. Let {(\Phi_n)} be the corresponding random walk on {Out(F)}. Then the limit

\displaystyle  \begin{array}{rcl}  \lim |\Phi_n(g)|^{1/n}=\lambda>1 \end{array}

exists almost always.

This is an analogue of Furstenberg’s theorem for {Sl(N,{\mathbb Z})}, and of Anders Karlsson for mapping class groups.

2. Oseledec type result

Here is a classical refinement of the above theorems.

Theorem 2 (Furstenberg-Kiefer, Hennion) There is a deterministic filtration {L_i} of {{\mathbb R}^N} and Lyapunov exponents {\lambda_i}

My version:

Theorem 3 There is a deterministic tree of subgroups {H} in {F=F_N} and Lyapunov exponents {\lambda_H} such that the growth has rate {\lambda_H} for elements of {F} conjugated into node {H} but in none of its children.

There are at most {\frac{3N-2}{4}} different positive Lyapunov exponents.

3. Horoboundary

This classical tool (Gromov ?) is used in the proof. Let {X} be a (possibly non symmetric) metric space. Map a point {x\in X} to the distance function, up to an additive constant. This maps {X} to {C(X)/{\mathbb R}}, equipped with the topology of uniform convergence on compact sets.

Proposition 4 (Walsh) Assume that {X} is geodesic, proper. Then The embedding is a homeomorphism onto its image, whose closure is compact.

Example. Horoboundary of the real line has 2 points.

We apply the following general fact to Outer space.

Theorem 5 (Karlsson-Ledrappier) Asymptotically, the growth of the distance to the origin of a random walk is modelled on the growth of a (random) horofunction {h}. I.e., if {(\Phi_n)} is a random walk on a discrete group acting isometrically on {X},

\displaystyle  \begin{array}{rcl}  \lim\frac{1}{n}d(x_0,\Phi_n^{-1}(x_0)=\lim\frac{-1}{n}h(\Phi_n^{-1}(x_0)). \end{array}

4. Outer space

On Outer space (the space of free actions of {F} on trees), we use the Francaviglia-Martino distance (Lipschitz distance). According to White, it is equal to the log of the supremal ratio of translation lengths. It is achieved by an element which is represented, on the quotient graph, by a simple loop, a figure 8 or a pair of glasses. In particular, it is a primitive element. This makes this distance handily computable.

The Ciller-Morgan compactification is obtained when mapping trees to their translation length, viewed as a function on {F}, up to rescaling. I modify this construction by restricting to primitive elements of {F}, getting what I call the primitive compactification. Elements in the closure are interpreted as isometric actions on {{\mathbb R}}-trees.

Theorem 6 The horocompactification of Outer space is homeomorphic to the primitive compactification. This in turn is a proper quotient of the Culler-Morgan compactification.

Example. If orbits of {F} on the real tree {T} are dense, then the equivalence class of {T} is reduced to {T}. But this us not always the case. Some equivalence classes are indeed non trivial

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Notes of Dominik Gruber’s lecture

Acylindrical hyperbolicity of graphical small cancellation groups

With Sisto.

We prove the theorem in the title and use it to exhibit new behaviours for the divergence function of a group.

Graphical small cancellation

It is an extension of small cancellation theory, devised by Gromov, in order to construct a finitely presented group weakly containing an expander. Gromov’s full construction uses (pseudo-)random choices, so the resulting presentation is not explicit. We shall not need these unpleasant steps, our presentations will be explicit.

Data: a graph {\Gamma}, edge orientations, edge labels in {S}. Consider the set of words read along closed paths. This is a normal subgroup of a free group, hence a quotient group {G(\Gamma)}.

By construction, {\Gamma} maps to {Cay(G(\Gamma),S)}. Need not be injective, unless we add assumptions: small cancellation.

A piece {p} is a labelled path that has at least two dustinct label-preserving maps to {\Gamma}. Say {\Gamma} (and {G(\Gamma)}) satisfies {C'(\lambda)} if ratios length of pieces over girth of {\Gamma} are {\leq \Gamma}.

Classical small cancellation amounts to {\Gamma} being a union of cycles. The language of van Kampen diagrams applies here as usual.

Theorem 1 (Gromov, Ollivier 2006) If {\Gamma} is a finite {C'(1/6)} graph, then {G(\Gamma)} is hyperbolic, and every component of {\Gamma} embeds isometrically into {Cay(G(\Gamma),S)}.

Theorem 2 (Gruber 2012) Let {\Gamma=\coprod_N \Gamma_n} is {C'(1/6)} graph, then {G(\Gamma)} is lacunary hyperbolic (i.e. at least one asymptotic cone is a real tree)

Theorem 3 (Gruber-Sisto) Let {\Gamma=\coprod_N \Gamma_n} is {C'(1/6)} graph, then {G(\Gamma)} is acylindrically hyperbolic.

1. Acylindrical hyperbolicity

See Hume and Sisto’s talks. The hyperbolic space {Y} on which {G(\Gamma)} acts is the Cayley graph of {G(\Gamma)} with respect to the (infinite) generating system consisting in {S} and the set of all words read along paths in {\Gamma}.

We use Strebel’s classification of geodesic triangles in {C'(1/2)} small cancellation groups. There are 7 cases, the hyperbolicity constant {\delta} is at most 2. Strebel’s argument goes through and shows that {Y} is 4-hyperbolic.

We show that all hyperbolic elements satisfy WPD. Thin quadrangles have width at most 2 in the middle.

2. Divergence

{Div(n)} measures the geodesic distance outside balls of radius {\leq n/2}. This a quasiisometry invariant. Examples with linear, quadratic, cubic, exponential divergence are known.

We show examples where the lim inf of {Div(n)/n^2} is 0, but the lim sup of {Div(n)/f(n)} is {+\infty} for every prescribed subexponential function.

We use large powers a WPD element to produce bridges that reduce divergence. Since, at every finite step of the construction, the group is hyperbolic, and this has exponential divergence, we may keep adding larger and larger relators to produce large (but subexponential) values of divergence.

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