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\title{Metric Geometry}
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\author{}
\begin{document}
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\section{Brownian Lamination \& Homeomorphism Theorem}
%In this tutorial, we introduce and study another ``representation" of the Continuum Brownian Tree. This is a sort of ``dual" of Aldous' CRT. We denote the open unit disk of $\mathbb{C}$ by $\mathbb{D}$ and its boundary by $\mathbb{S}_{1}$.

\begin{exo}[Brownian Lamination] Let $\mathbf{e}:[0,1] \longrightarrow [0,+\infty)$ be a normalized Brownian excursion. Recall that we proved that the local minima of $\mathbf{e}$ are almost surely distinct. We associate a random closed subset of the closed unit disk $\overline{\mathbb{D}}$ to $\mathbf{e}$ by the following device. Recall the definition of the pseudo-distance associated to $\mathbf{e}$: for $a,b \in [0,1]$ we put
$$ {d}_{\mathbf{e}}(a,b) = \mathbf{e}(a)+\mathbf{e}(b) - 2 \inf \big\{\mathbf{e}(u) : u \in [a \wedge b, a \vee b]\big\}.$$
Let $L_{\mathbf{e}}$ be the union of all segments $[xy]$ where $x = \exp(2i\pi a)$ and $y=\exp(2i\pi b)$ with ${d}_{\mathbf{e}}(a,b)=0$. A segment $[xy]$ with endpoints on $\mathbb{S}_{1}$ is called a \emph{chord}. If two chords $[xy]$ and $[x'y']$ are such that $(xy)\cap(x'y')=\varnothing$ we say that the chords are \emph{non-crossing}. A closed subset of $\overline{\mathbb{D}}$ which can be written as a union of non-crossing chords is called a \emph{lamination}.
\begin{enumerate}
\item Show that $L_{\mathbf{e}}$ is a closed subset of $\overline{\mathbb{D}}$.
\item Show that a.s.\,if $a,b,c,d \in [0,1]$ such that ${d}_{\mathbf{e}}(a,b)={d}_{\mathbf{e}}(c,d)=0$ then 
$$ \mbox{ either } [e^{2i\pi a}e^{2i\pi b}]= [e^{2i\pi c}e^{2i\pi d}] \hspace{0.3cm}\mbox{ or }\hspace{0.3cm} (e^{2i\pi a}e^{2i\pi b})\cap(e^{2i\pi c}e^{2i\pi d})= \varnothing.$$ Conclude that a.s.\,$L_{\mathbf{e}}$ is lamination.
\item Show that a.s.\,the connected components of $\overline{\mathbb{D}}\backslash L_{\mathbf{e}}$ are open triangles with vertices on $\mathbb{S}_{1}$.
\item Show that a.s.\,$L_{\mathbf{e}}$ is maximal for the inclusion relation among laminations.
\end{enumerate}
We define a relation on $\overline{\mathbb{D}}$ using $\mathbf{e}$: if $x,y \in \overline{\mathbb{D}}$, we put $x \sim_{\mathbf{e}}y$  if $x$ and $y$ belong to a chord $[e^{2i\pi a}e^{2i\pi b}]$ with ${d}_{\mathbf{e}}(a,b)=0$ or if $x$ and $y$ belong to the closure of some open triangle of $\overline{\mathbb{D}} \backslash L_{\mathbf{e}}$. 


\begin{enumerate} 
\setcounter{enumi}{4}
\item Prove that $\sim_\mathbf{e}$ is a closed equivalence relation and that the quotient space $\overline{\mathbb{D}} / \sim_{\mathbf{e}}$ is homeomorphic to the $\R$-tree $T_{\mathbf{e}}$ coded by $\mathbf{e}$.

  \small 

 \noindent \textbf{Reminder on quotient topology:} If $X$ is topological space and $\sim$ an equivalence relation on $X$, we endow $X/\sim$ with the finest topology for which the canonical projection $\pi: X \to X/\sim$ is continuous. Equivalently, a set $A \subset X /\sim$ is open if and only if $ \pi^{-1}(A)$ is open. We say that $\sim$ is closed if the set $\{(x,y) \in X\times X : x \sim y \},$ is closed. We admit (or we prove) that if $X$ is a compact metric space and $\sim$ is closed then $X/\sim$ is an Hausdorff space and then compact. \normalsize 
 \item Show that the local minima of $\mathbf{e}$ are dense in $[0,1]$ and deduce that $L_{\mathbf{e}}$ has an empty interior.
%\item * Any guess about the Hausdorff dimension of $L_{\mathbf{e}}$ ? Draw a picture of $L_{\mathbf{e}}$. 
\end{enumerate}
\end{exo}

\begin{exo}[Homeomorphism Theorem] We now consider, together with $\mathbf{e}$, the Head of the Brownian snake $Z$ driven by $\mathbf{e}$. We can do exactly the same procedure for $Z$ (in particular we admit that the local minima of $Z$ are distinct). Thus $Z$ furnishes an equivalence relation $\sim_{Z}$ on $\overline{\mathbb{D}}$ is a similar manner as to $\mathbf{e}$. We consider $\mathbb{S}_{2}$ the standard Euclidean sphere of radius $1$ in $\mathbb{R}^3$ and put $$H_{+} = \big\{ (x_{1},x_{2},x_{3}) \in \mathbb{R}^3,x_{1}^2+x_{2}^2+x_{3}^2=1 \mbox{ and } x_{3} \geq 0 \big\},$$ for the closed North hemisphere of $\mathbb{S}_{2}$ and similarly $H_{-}$ denotes the closed South hemisphere. The stereographic projections from the North and South poles enable us to identify $H_{+}$ and $H_{-}$ with $\overline{\mathbb{D}}$. We will associate the function $\mathbf{e}$ (resp.\,$Z$) to the North (resp.\,South) part of the ball, hence we can define $\sim_{\mathbf{e}}$ on $H_{+}$ and $\sim_{Z}$ on $H_{-}$.
\begin{enumerate}
\item Check that $H_{+}/\sim_{\mathbf{e}}$ is still homeomorphic to $T_{\mathbf{e}}$.
\end{enumerate}
We put a relation on $x,y\in \mathbb{S}_{2}$ by $x \sim y$ if and only if $x,y \in H_{+}$ and $ x\sim_{\mathbf{e}} y$ or $x,y \in H_{-}$ and $x\sim_{Z}y$. We admit the following  fact about the process $(\mathbf{e}_{t},Z_{t})_{t \in[0,1]}$. Almost surely, for every $s \in\,]0,1[$ such that for some $\varepsilon>0$ if we have $$\mathbf{e}_{s}= \min_{r \in [s-\varepsilon,s]}\mathbf{e}_{r}\mbox{\ \ \  or \ \ \ } \mathbf{e}_{s}= \min_{r \in [s,s+\varepsilon]}\mathbf{e}_{r}$$  then $$Z_{s}> \min_{r \in [s-\delta,s]}Z_{r}\mbox{, for every }0<\delta <s \mbox{\ \ \ and \ \ \ } Z_{s}> \min_{r \in [s,s+\delta]}Z_{r}\mbox{, for every }0<\delta <1-s.$$


 \begin{enumerate}
\setcounter{enumi}{1}
\item Prove that a.s.\,$\sim$ is a closed equivalence relation.
\end{enumerate}
\begin{theorem}[Moore (1925)] Let $\sim$ be a closed equivalence relation on the two dimensional sphere $\mathbb{S}_{2}$. Assume that every equivalence class of $\sim$ is a compact path-connected subset of the sphere whose complement is connected. The quotient space $\mathbb{S}_{2}/\sim$ is homeomorphic to $\mathbb{S}_{2}$.
\end{theorem}
\begin{enumerate}
\setcounter{enumi}{2}
\item Give an example of a closed equivalence relation $\simeq$ such that the quotient $\mathbb{S}_{2}/ \simeq$ is not homeomorphic to $\mathbb{S}_{2}$.
\item Prove that in our setting $\sim$ a.s.\,verifies all hypotheses of Moore's Theorem and deduce that almost surely $\mathbb{S}_2 / \sim$ is homeomorphic to $\mathbb{S}_2$.
\item Does anybody see a link with scaling limits of random planar quadrangulations ?
\end{enumerate}
\end{exo}



\begin{exo}Who is this charming gentleman / What does represent these nice pictures ?
\begin{figure}[h] 
\begin{center} 
\includegraphics[height=4cm]{thurston}
\hspace{0.5cm}
\includegraphics[height=4cm]{fig1}
\hspace{0.5cm}
\includegraphics[height=4cm]{thurstonbisbis}
 \end{center}
 The third picture is taken from Thurston. \end{figure}
\end{exo}
\nocite{LGP08}
\nocite{Ald94a}
\nocite{Ald94b}\bibliographystyle{alpha}
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