\end{array}
\right.$$
Their topological disorder can then be studied.
+To do so, a relevant distance must be defined of $\mathcal{X}$, as
+follows~\cite{GuyeuxThese10,bgw09:ip}:
+$$d((S,E);(\check{S};\check{E})) = d_e(E,\check{E}) + d_s(S,\check{S})$$
+\noindent where $\displaystyle{d_e(E,\check{E}) = \sum_{k=1}^\mathsf{N} \delta (E_k, \check{E}_k)}$, ~~and~ $\displaystyle{d_s(S,\check{S}) = \dfrac{9}{\textsf{N}} \sum_{k = 1}^\infty \dfrac{|S^k-\check{S}^k|}{10^k}}$.
+This new distance has been introduced in \cite{bgw09:ip} to satisfy the following requirements.
+\begin{itemize}
+\item When the number of different cells between two systems is increasing, then their distance should increase too.
+\item In addition, if two systems present the same cells and their respective strategies start with the same terms, then the distance between these two points must be small because the evolution of the two systems will be the same for a while. Indeed, the two dynamical systems start with the same initial condition, use the same update function, and as strategies are the same for a while, then components that are updated are the same too.
+\end{itemize}
+The distance presented above follows these recommendations. Indeed, if the floor value $\lfloor d(X,Y)\rfloor $ is equal to $n$, then the systems $E, \check{E}$ differ in $n$ cells. In addition, $d(X,Y) - \lfloor d(X,Y) \rfloor $ is a measure of the differences between strategies $S$ and $\check{S}$. More precisely, this floating part is less than $10^{-k}$ if and only if the first $k$ terms of the two strategies are equal. Moreover, if the $k^{th}$ digit is nonzero, then the $k^{th}$ terms of the two strategies are different.
+It has then be stated that
+\begin{proposition}
+$G_f : (\mathcal{X},d) \to (\mathcal{X},d)$ is a continuous function
+\end{proposition}
-
-
-%\frame{
-% \frametitle{Métrique et continuité}
-
-%Distance sur $\mathcal{X}:$
-%$$d((S,E);(\check{S};\check{E})) = d_e(E,\check{E}) + d_s(S,\check{S})$$
-
-%\noindent où $\displaystyle{d_e(E,\check{E}) = \sum_{k=1}^\mathsf{N} \delta (E_k, \check{E}_k)}$, ~~et~ $\displaystyle{d_s(S,\check{S}) = \dfrac{9}{\textsf{N}} \sum_{k = 1}^\infty \dfrac{|S^k-\check{S}^k|}{10^k}}$.
-%%\end{block}
-
-%\vspace{0.5cm}
-
-%\begin{alertblock}{Théorème}
-%La fonction $G_f : (\mathcal{X},d) \to (\mathcal{X},d)$ est continue.
-%\end{alertblock}
-
-%}
-
-
+With all this material, the study of chaotic iterations as a discrete
+dynamical system has then be realized.
+This study is summarized in the next section.
% \frame{
% \frametitle{\'Etude de $(\mathcal{X},d)$}
