\section{Introduction}
-\section{Topology}
+\section{Topologycal Study of Disorder}
-\subsection{Historical context}
+\subsection{Historical Context}
Pseudorandom number generators are recurrent sequences having a disordered behavior.
$f:I \longrightarrow I$ continuous on the line segment $I$, the absence of
any 2-cycle implies the convergence of the discrete dynamical system.
+This theorem establish a clear link between the existence of a cycle of
+a given length and the convergence of the system. In other words, between
+cycles and order. Conversely, Li and Yorke have established in 1975~\cite{Li75} that
+the presence of a point of period three implies chaos in the same situation
+than previously. By chaos, they mean the existence of points of any
+period: this kind of disorder, which is the first occurrence of the
+term ``chaos'' in the mathematical litterature, is thus related to the
+multiplicity of periods. Since that time, the mathematical theory of
+chaos has known several developments to qualify or quantify the richness
+of chaos presented by a given discrete dynamical system, one of the most
+famous work, although old, being the one of Devaney~\cite{devaney}.
+
+\subsection{Iterative Systems}
+
+In the distributed computing community, dynamical systems have been
+generatized to take into account delay transmission or heterogeneous
+computational powers. Mathematically, the intended result is often one
+fixed point resulting from the iterations of a given function over a
+Boolean vector, considering that:
+\begin{itemize}
+\item at time $t$, $x^{t}$ is computed using not only $x^{t-1}$, but
+potentially any $x^{k}, k<t$, due to delay transmission,
+\item not all the components of $x^{t}$ are supposed to be updated at
+each iteration: each component represents a unit of computation, and
+these units have not the same processing frequency.
+\end{itemize}
+
+These considerations lead to the following definition of an iterative
+system.
+
+\begin{definition}
+Iterative systems on a set $\mathcal{X}$ are defined by
+$$\left\{
+ \begin{array}{l}
+ x^0 \in \mathcal{X}\\
+ x^{n+1} = f^n(x^0, \hdots, x^n)
+ \end{array}
+ \right.$$
+where $f^n:\mathcal{X}^{n+1}\rightarrow \mathcal{X}$.
+\end{definition}
-%
-%
-% \begin{block}{Convergence}
-% \begin{itemize}
-% \item $f$ monotone
-% \item Applications contractantes
-% \item Coppel: Pas de 2-cycle $\Rightarrow$ convergence
-% \end{itemize}
-% \end{block}
-%}
-%\frame{
-% \frametitle{3-cycle implique chaos}
-% \begin{alertblock}{Period Three Implies Chaos (Li et Yorke, 1975)}
-%S'il y a un point de période 3, alors il y a un point de n'importe quelle période
-% \end{alertblock}
-%
-% \uncover<2->{
-% \begin{exampleblock}{Remarques}
-% \begin{itemize}
-% \item Désordre lié à la multiplicité des périodes
-% \item \`A AND, on étudie des ``systèmes itératifs'' pour le calcul distribué, généralisation des suites récurrentes
-% \end{itemize}
-% \end{exampleblock}
-% }
-%}
-
-
-
-
-%%\subsection*{Réécriture des systèmes itératifs}
-
-%%\frame{
-%% \frametitle{Les systèmes itératifs: généralisation}
-%% \begin{block}{Les systèmes itératifs en toute généralité}
-%% La formulation suivante englobe tous les modes d'itérations imaginables:
-%% $$\left\{
-%% \begin{array}{l}
-%% x^0 \in \mathcal{X}\\
-%% x^{n+1} = f^n(x^0, \hdots, x^n)
-%% \end{array}
-%% \right.$$
-%% où $f^n:\mathcal{X}^{n+1}\rightarrow \mathcal{X}$
%% \end{block}
%%\uncover<2->{
%%Différents modes d'itérations: séries, parallèles, chaotiques, asynchrones...
