+Obviously, this exponent must be positive to have a multiplication of
+initial errors, and thus chaos in this understanding.
+
+Having all these definitions in mind, we have studied the topological
+behavior of chaotic iterations presented in Definition~\ref{defIC}.
+
+\section{The Study of Iterative Systems}
+
+We have firstly stated that~\cite{gb11:bc,GuyeuxThese10}:
+
+\begin{theorem}
+ $G_{f_0}$ is regular and transitive on $(\mathcal{X},d)$, thus it is
+ chaotic according to Devaney.
+Furthermore, its constant of sensibility is greater than $\mathsf{N}-1$.
+\end{theorem}
+
+Thus the set $\mathcal{C}$ of functions $f:\mathds{B}^\mathsf{N}
+\longrightarrow \mathds{B}^\mathsf{N}$ making the chaotic iterations of
+Definition~\ref{defIC} a case of chaos according to Devaney, is a nonempty
+set. To characterize functions of $\mathcal{C}$, we have firstly stated
+that transitivity implies regularity for these particular iterated
+systems~\cite{bcgr11:ip}. To achieve characterization, we then have introduced the
+following graph.
+
+\begin{figure}
+ \centering
+ \includegraphics[scale=0.55]{grapheTPICver2.pdf}
+ \caption{Example of an asynchronous iteration graph}
+ \label{GTPIC}
+ \end{figure}
+
+
+
+Let $f$ be a map from $\mathds{B}^\mathsf{N}$ to itself. The
+{\emph{asynchronous iteration graph}} associated with $f$ is the
+directed graph $\Gamma(f)$ defined by: the set of vertices is
+$\mathds{B}^\mathsf{N}$; for all $x\in\mathds{B}^\mathsf{N}$ and
+$i\in \llbracket1;\mathsf{N}\rrbracket$,
+the graph $\Gamma(f)$ contains an arc from $x$ to $F_f(i,x)$.
+The relation between $\Gamma(f)$ and $G_f$ is clear: there exists a
+path from $x$ to $x'$ in $\Gamma(f)$ if and only if there exists a
+strategy $s$ such that the parallel iteration of $G_f$ from the
+initial point $(s,x)$ reaches the point $x'$. Figure~\ref{GTPIC} presents
+such an asynchronous iteration graph.
+We thus have proven that~\cite{bcgr11:ip}.
+
+
+\begin{theorem}
+$G_f$ is transitive, and thus chaotic according to Devaney,
+if and only if $\Gamma(f)$ is strongly connected.
+\end{theorem}
+
+This characterization makes it possible to quantify the number of
+functions in $\mathcal{C}$: it is equal to
+ $\left(2^\mathsf{N}\right)^{2^\mathsf{N}}$.
+Then the study of the topological properties of disorder of these
+iterative systems has been further investigated, leading to the following
+results.
+
+\begin{theorem}
+ $\forall f \in \mathcal{C}$, $Per\left(G_f\right)$ is infinitely countable, $G_f$ is strongly transitive and is chaotic according to Knudsen. It is thus undecomposable, unstable, and chaotic as defined by Wiggins.
+ \end{theorem}
+
+ \begin{theorem}
+$\left(\mathcal{X}, G_{f_0}\right)$ is topologically mixing,
+ expansive (with a constant equal to 1), chaotic as defined by
+ Li and Yorke, and has a topological entropy and an exponent of Lyapunov
+ both equal to $ln(\mathsf{N})$.
+\end{theorem}
+
+At this stage, a new kind of iterative systems that only manipulates
+integers have been discovered, leading to the questioning of their
+computing for security applications. In order to do so, the possibility
+of their computation without any loss of chaotic properties has first
+been investigated. These chaotic machines are presented in the next
+section.
+
+
+
+
+\section{Topology of Programs}
+
+
+The next stage was to prove that chaos was possible in finite machine.
+The two main problems were that: (1) Chaotic sequences are usually
+defined in the real line whereas define real numbers on computers
+is impossible. (2) All finite state machines always enter into a cycle
+when iterating, and this periodic behavior cannot be stated as chaotic.
+
+The first problem is disputable, as the shadow lemma proves that, when
+considering the sequence $x^{n+1} = trunc_k\left(f(x^n)\right)$, where
+$(f,[0,1])$ is a chaotic dynamical system and $trunc_k(x) =
+\dfrac{\lfloor 10^k x \rfloor}{10^k}$ is the truncated version of
+$x\in \mathds{R}$ at its $k-$th digits, then the sequence $(x^n)$ is as
+close as possible to a real chaotic orbit. Thus iterating a chaotic
+function on floating point numbers does not deflate the chaotic behavior
+as much. However, even if this first claim is not really a problem, we
+have prevent from any disputation by considering a tool (the chaotic
+iterations) that only manipulates integers bounded by $\mathsf{N}$.
+
+The second claim is surprisingly never considered as an issue when
+considering the generation of randomness on computers.
