+\emph{A priori}, there is no relation between these chaotic iterations
+and the mathematical theory of chaos recalled in the previous section.
+On our side, we have regarded whether these chaotic iterations can
+behave chaotically, as it is defined for instance by Devaney, and if so,
+in which applicative context this behavior can be profitable.
+This questioning has led to a first necessary condition of non convergence~\cite{GuyeuxThese10}.
+
+\begin{proposition}
+Let $f : \mathds{B}^\mathsf{N} \to \mathds{B}^\mathsf{N}$ and
+$S \in \llbracket 1, \mathsf{N} \rrbracket^{\mathds{N}}$.
+If the chaotic iterations $\left(f,(x^0,S)\right)$ are not convergent, then:
+\begin{itemize}
+\item either $f$ is not a contraction, meaning that there is no Boolean matrix
+ $M$ of size $\mathsf{N}$ satisfying $\forall x,y\in \mathds{B}^\mathsf{N}$,
+ $d(f(x),f(y)) \leqslant M d(x,y)$, where $d$ is here the ``vectorial distance''
+ defined by $d(x,y) = \left(\begin{array}{c} \delta(x_1,y_1)\\ \vdots \\ \delta(x_\mathsf{N},
+ y_\mathsf{N}) \end{array}\right)$, with $\delta$ the discrete metric defined by $\delta(x,y) = \left\{\begin{matrix} 1 &\mbox{if}\ x\neq y , \\ 0 &\mbox{if}\ x = y \end{matrix}\right.$, and $\leqslant$ is the inequality term by term~\cite{Robert}.
+\item or $S$ is not pseudo-periodic: it is not constituted by an infinite succession of finite sequences, each having any element of $\llbracket
+ 1, \mathsf{N} \rrbracket$ at least once.
+\end{itemize}
+\end{proposition}
+
+The second alternative of the proposition above concerns the strategy,
+which should be provided by the outside world. Indeed, in our opinion,
+chaotic iterations can receive a PRNG $S$ as input, and due to
+properties of disorder of $f$, generate a new pseudorandom sequence
+that presents better statistical properties than $S$. Having this
+approach in mind, we thus have searched vectorial Boolean iteration
+functions that are not contractions. The vectorial negation function
+$f_0:\mathds{B}^\mathsf{N} \longrightarrow \mathds{N}^\mathsf{N},$
+$(x_1, \hdots, x_\mathsf{N}) \longmapsto (\overline{x_1}, \hdots,
+\overline{x_\mathsf{N}}) $ is such a function, which served has a
+model in our further studies ($\overline{x}$ stands for the negation
+of the Boolean $x$).
+
+The quantity of disorder generated by such chaotic iterations, when
+satisfying the proposition above, has then been measured. To do so,
+chaotic iterations have first been rewritten as simple discrete
+dynamical systems, as follows.
+
+
+\subsection{Chaotic Iterations as Dynamical Systems}
+
+The problems raised by such a formalization can be summarized as
+follows.
+Chaotic iterations are defined in the discrete mathematics framework,
+considering $x^0 \in \mathds{B}^\mathds{N}$ and $S \in \mathcal{S} = \llbracket 1,\mathsf{N}\rrbracket^\mathds{N}$, and iterations having the
+form
+$$x_i^{n+1} = \left\{ \begin{array}{ll} x^{n}_{i} & \textrm{ si } i \neq S^n\\ f(x^{n})_{i} & \textrm{ si } i = S^n \end{array} \right.$$
+where $f: \mathds{B}^\mathsf{N} \to \mathds{B}^\mathsf{N}$.
+However, the mathematical theory of chaos takes place into a
+topological space $(\mathcal{X},\tau)$. It studies the iterations
+$x^0 \in \mathcal{X}$, $\forall n \in \mathds{N}, x^{n+1} = f(x^n)$,
+where $f : \mathcal{X} \to \mathcal{X}$ is continuous for the
+topology $\tau$.
+
+To realize the junction between these two frameworks, the following
+material has been introduced~\cite{GuyeuxThese10,bgw09:ip}:
+\begin{itemize}
+\item the shift function: $\sigma : \mathcal{S} \longrightarrow \mathcal{S}, (S^n)_{n \in \mathds{N}} \mapsto (S^{n+1})_{n \in \mathds{N}}$.
+\item the initial function, defined by $i : \mathcal{S} \longrightarrow \llbracket 1 ; \mathsf{N} \rrbracket, (S^n)_{n \in \mathds{N}} \mapsto S^0$
+\item and $F_f : \llbracket 1 ; \mathsf{N} \rrbracket \times \mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N},$ $$(k,E) \longmapsto \left( E_j.\delta(k,j) + f(E)_k.\overline{\delta (k,j)} \right)_{j \in \llbracket 1 ; \mathsf{N} \rrbracket}$$
+\end{itemize}
+where $\delta$ is the discrete metric.
