\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})$$
+\noindent where $\displaystyle{d_e(E,\check{E}) = \sum_{k=1}^\mathsf{N} \delta (E_k, \check{E}_k)}$, ~~and~ $\displaystyle{d_s(S,\check{S}) = \dfrac{9}{\textsf{N}} \sum_{k = 1}^\infty \dfrac{|S^k-\check{S}^k|}{10^k}}$.
+This new distance has been introduced in \cite{bgw09:ip} to satisfy the following requirements.
+\begin{itemize}
+\item When the number of different cells between two systems is increasing, then their distance should increase too.
+\item In addition, if two systems present the same cells and their respective strategies start with the same terms, then the distance between these two points must be small because the evolution of the two systems will be the same for a while. Indeed, the two dynamical systems start with the same initial condition, use the same update function, and as strategies are the same for a while, then components that are updated are the same too.
+\end{itemize}
+The distance presented above follows these recommendations. Indeed, if the floor value $\lfloor d(X,Y)\rfloor $ is equal to $n$, then the systems $E, \check{E}$ differ in $n$ cells. In addition, $d(X,Y) - \lfloor d(X,Y) \rfloor $ is a measure of the differences between strategies $S$ and $\check{S}$. More precisely, this floating part is less than $10^{-k}$ if and only if the first $k$ terms of the two strategies are equal. Moreover, if the $k^{th}$ digit is nonzero, then the $k^{th}$ terms of the two strategies are different.
+It has then be stated that
+\begin{proposition}
+$G_f : (\mathcal{X},d) \to (\mathcal{X},d)$ is a continuous function
+\end{proposition}
+With all this material, the study of chaotic iterations as a discrete
+dynamical system has then be realized.
+This study is summarized in the next section.
+\subsection{Topological Properties of Chaotic Iterations}
-%\frame{
-% \frametitle{Métrique et continuité}
-
-%Distance sur $\mathcal{X}:$
-%$$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}
-
-%}
-
-
-
-% \frame{
-% \frametitle{\'Etude de $(\mathcal{X},d)$}
-% \begin{block}{Propriétés de $(\mathcal{X},d)$}
-% \begin{itemize}
-% \item $\mathcal{X}$ est infini indénombrable
-% \vspace{0.15cm}
-% \item $(\mathcal{X},d)$ est un espace métrique compact, complet et parfait
-% \end{itemize}
-% \end{block}
-%
-% \vspace{0.5cm}
-%
-% \begin{block}{\'Etude de $G_{f_0}$}
-% $G_{f_0}$ est surjective, mais pas injective \vspace{0.3cm}\newline $\Rightarrow (\mathcal{X},G_{f_0})$ pas réversible.
-% \end{block}
-
-% }
-
+The topological space on which chaotic iterations are defined has
+firstly been investigated, leading to the following result~\cite{gb11:bc,GuyeuxThese10}:
+\begin{proposition}
+$\mathcal{X}$ is an infinitely countable metric space, being both
+compact, complete, and perfect (each point is an accumulation point).
+\end{proposition}
+These properties are required in some topological specific
+formalization of a chaotic dynamical system, justifying their
+proofs.
+Concerning $G_{f_0}$, it has been stated that~\cite{GuyeuxThese10}.
+\begin{proposition}
+$G_{f_0}$ is surjective, but not injective, and so the dynamical system $(\mathcal{X},G_{f_0})$ is not reversible.
-%%\frame{
-%% \frametitle{Etude des périodes}
-%% \begin{block}{Multiplicité des périodes ?}
-%% Soit $f_0:\mathds{B}^\mathsf{N} \rightarrow \mathds{B}^\mathsf{N}$ la négation vectorielle.
-%% \begin{itemize}
-%% \item $\forall k \in \mathds{N}, Per_{2k+1}(G_{f_0}) = \varnothing, card\left(Per_{2k+2}(G_{f_0})\right)>0$ \vspace{0.3cm} \linebreak $\Rightarrow G_{f_0}$ pas chaotique sur $\mathcal{X}$
-%% \item Cependant :
-%% \begin{itemize}
-%% \item Il y a chaos sur $\mathcal{X}^G = \mathcal{P}\left(\llbracket 1,\mathsf{N}\rrbracket\right)^\mathds{N}\times \mathds{B}^\mathsf{N}$.
-%% \item $G_{f_0}$ possède plus de $n^2$ points périodiques de période $2n$.
-%% \end{itemize}
-%% \end{itemize}
-%% \end{block}
-%% \uncover<2->{
-%% Cette multiplicité des périodes n'est pas le désordre complet...
-%% }
-%%}
+Furthermore, if we denote by $Per_k(f)$ the set of periodic points
+of period $k$ for $f$, we have
+ $\forall k \in \mathds{N}, Per_{2k+1}(G_{f_0}) = \varnothing, card\left(Per_{2k+2}(G_{f_0})\right)>0$.
+\end{proposition}
+
+So $\Rightarrow G_{f_0}$ does not present the existence of points of any period referred as chaos in the article of Li and Yorke~\cite{Li75}.
+However~\cite{GuyeuxThese10}:
+ \begin{itemize}
+ \item This kind of disorder can be stated on $\mathcal{X}^G = \mathcal{P}\left(\llbracket 1,\mathsf{N}\rrbracket\right)^\mathds{N}\times \mathds{B}^\mathsf{N}$.
+ \item $G_{f_0}$ possesses more than $n^2$ points of period $2n$.
+ \end{itemize}
+Additionally, this existence of points of any period has been rejected
+by the community to the benefit of more recent notions of chaos, as
+they are detailed in the following paragraphs.
