\usepackage{amscd}
\usepackage{moreverb}
\usepackage{commath}
+\usepackage{algorithm2e}
\usepackage[standard]{ntheorem}
% Pour mathds : les ensembles IR, IN, etc.
\title{Efficient generation of pseudo random numbers based on chaotic iterations on GPU}
\begin{document}
+
+\author{Jacques M. Bahi, Rapha\"{e}l Couturier, and Christophe Guyeux\thanks{Authors in alphabetic order}}
+
\maketitle
\begin{abstract}
Interet de générer des nombres alea sur GPU
...
-\section{Chaotic iterations}
+% >>>>>>>>>>>>>>>>>>>>>> Basic recalls <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
+\section{Basic Recalls}
+\label{section:BASIC RECALLS}
+This section is devoted to basic definitions and terminologies in the fields of topological chaos and chaotic iterations.
+\subsection{Devaney's chaotic dynamical systems}
+
+In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$ denotes the $i^{th}$ component of a vector $V$. $f^{k}=f\circ ...\circ f$ denotes the $k^{th}$ composition of a function $f$. Finally, the following notation is used: $\llbracket1;N\rrbracket=\{1,2,\hdots,N\}$.
+
+
+Consider a topological space $(\mathcal{X},\tau)$ and a continuous function $f : \mathcal{X} \rightarrow \mathcal{X}$.
+
+\begin{definition}
+$f$ is said to be \emph{topologically transitive} if, for any pair of open sets $U,V \subset \mathcal{X}$, there exists $k>0$ such that $f^k(U) \cap V \neq \varnothing$.
+\end{definition}
+
+\begin{definition}
+An element $x$ is a \emph{periodic point} for $f$ of period $n\in \mathds{N}^*$ if $f^{n}(x)=x$.% The set of periodic points of $f$ is denoted $Per(f).$
+\end{definition}
+
+\begin{definition}
+$f$ is said to be \emph{regular} on $(\mathcal{X}, \tau)$ if the set of periodic points for $f$ is dense in $\mathcal{X}$: for any point $x$ in $\mathcal{X}$, any neighborhood of $x$ contains at least one periodic point (without necessarily the same period).
+\end{definition}
+
+
+\begin{definition}
+$f$ is said to be \emph{chaotic} on $(\mathcal{X},\tau)$ if $f$ is regular and topologically transitive.
+\end{definition}
+
+The chaos property is strongly linked to the notion of ``sensitivity'', defined on a metric space $(\mathcal{X},d)$ by:
+
+\begin{definition}
+\label{sensitivity} $f$ has \emph{sensitive dependence on initial conditions}
+if there exists $\delta >0$ such that, for any $x\in \mathcal{X}$ and any neighborhood $V$ of $x$, there exist $y\in V$ and $n > 0$ such that $d\left(f^{n}(x), f^{n}(y)\right) >\delta $.
+
+$\delta$ is called the \emph{constant of sensitivity} of $f$.
+\end{definition}
+
+Indeed, Banks \emph{et al.} have proven in~\cite{Banks92} that when $f$ is chaotic and $(\mathcal{X}, d)$ is a metric space, then $f$ has the property of sensitive dependence on initial conditions (this property was formerly an element of the definition of chaos). To sum up, quoting Devaney in~\cite{Devaney}, a chaotic dynamical system ``is unpredictable because of the sensitive dependence on initial conditions. It cannot be broken down or simplified into two subsystems which do not interact because of topological transitivity. And in the midst of this random behavior, we nevertheless have an element of regularity''. Fundamentally different behaviors are consequently possible and occur in an unpredictable way.
+
+
+
+\subsection{Chaotic iterations}
+\label{sec:chaotic iterations}
+
+
+Let us consider a \emph{system} with a finite number $\mathsf{N} \in
+\mathds{N}^*$ of elements (or \emph{cells}), so that each cell has a
+Boolean \emph{state}. Having $\mathsf{N}$ Boolean values for these
+ cells leads to the definition of a particular \emph{state of the
+system}. A sequence which elements belong to $\llbracket 1;\mathsf{N}
+\rrbracket $ is called a \emph{strategy}. The set of all strategies is
+denoted by $\mathbb{S}.$
+
+\begin{definition}
+\label{Def:chaotic iterations}
+The set $\mathds{B}$ denoting $\{0,1\}$, let
+$f:\mathds{B}^{\mathsf{N}}\longrightarrow \mathds{B}^{\mathsf{N}}$ be
+a function and $S\in \mathbb{S}$ be a strategy. The so-called
+\emph{chaotic iterations} are defined by $x^0\in
+\mathds{B}^{\mathsf{N}}$ and
+$$
+\forall n\in \mathds{N}^{\ast }, \forall i\in
+\llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{
+\begin{array}{ll}
+ x_i^{n-1} & \text{ if }S^n\neq i \\
+ \left(f(x^{n-1})\right)_{S^n} & \text{ if }S^n=i.
+\end{array}\right.
+$$
+\end{definition}
+
+In other words, at the $n^{th}$ iteration, only the $S^{n}-$th cell is
+\textquotedblleft iterated\textquotedblright . Note that in a more
+general formulation, $S^n$ can be a subset of components and
+$\left(f(x^{n-1})\right)_{S^{n}}$ can be replaced by
+$\left(f(x^{k})\right)_{S^{n}}$, where $k<n$, describing for example,
+delays transmission~\cite{Robert1986,guyeux10}. Finally, let us remark that
+the term ``chaotic'', in the name of these iterations, has \emph{a
+priori} no link with the mathematical theory of chaos, recalled above.
+
+
+Let us now recall how to define a suitable metric space where chaotic iterations are continuous. For further explanations, see, e.g., \cite{guyeux10}.
+
+Let $\delta $ be the \emph{discrete Boolean metric}, $\delta (x,y)=0\Leftrightarrow x=y.$ Given a function $f$, define the function:
+\begin{equation*}
+\begin{array}{lrll}
+F_{f}: & \llbracket1;\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 \llbracket1;\mathsf{N}\rrbracket},%
+\end{array}%
+\end{equation*}%
+\noindent where + and . are the Boolean addition and product operations.
+Consider the phase space:
+\begin{equation*}
+\mathcal{X} = \llbracket 1 ; \mathsf{N} \rrbracket^\mathds{N} \times
+\mathds{B}^\mathsf{N},
+\end{equation*}
+\noindent and the map defined on $\mathcal{X}$:
+\begin{equation}
+G_f\left(S,E\right) = \left(\sigma(S), F_f(i(S),E)\right), \label{Gf}
+\end{equation}
+\noindent where $\sigma$ is the \emph{shift} function defined by $\sigma (S^{n})_{n\in \mathds{N}}\in \mathbb{S}\longrightarrow (S^{n+1})_{n\in \mathds{N}}\in \mathbb{S}$ and $i$ is the \emph{initial function} $i:(S^{n})_{n\in \mathds{N}} \in \mathbb{S}\longrightarrow S^{0}\in \llbracket 1;\mathsf{N}\rrbracket$. Then the chaotic iterations defined in (\ref{sec:chaotic iterations}) can be described by the following iterations:
+\begin{equation*}
+\left\{
+\begin{array}{l}
+X^0 \in \mathcal{X} \\
+X^{k+1}=G_{f}(X^k).%
+\end{array}%
+\right.
+\end{equation*}%
+
+With this formulation, a shift function appears as a component of chaotic iterations. The shift function is a famous example of a chaotic map~\cite{Devaney} but its presence is not sufficient enough to claim $G_f$ as chaotic.
+
+To study this claim, a new distance between two points $X = (S,E), Y = (\check{S},\check{E})\in
+\mathcal{X}$ has been introduced in \cite{guyeux10} as follows:
+\begin{equation*}
+d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}),
+\end{equation*}
+\noindent where
+\begin{equation*}
+\left\{
+\begin{array}{lll}
+\displaystyle{d_{e}(E,\check{E})} & = & \displaystyle{\sum_{k=1}^{\mathsf{N}%
+}\delta (E_{k},\check{E}_{k})}, \\
+\displaystyle{d_{s}(S,\check{S})} & = & \displaystyle{\dfrac{9}{\mathsf{N}}%
+\sum_{k=1}^{\infty }\dfrac{|S^k-\check{S}^k|}{10^{k}}}.%
+\end{array}%
+\right.
+\end{equation*}
+
+
+This new distance has been introduced 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.
+
+Finally, it has been established in \cite{guyeux10} that,
+
+\begin{proposition}
+$G_{f}$ must be continuous in the metric
+space $(\mathcal{X},d)$.
+\end{proposition}
+
+The chaotic property of $G_f$ has been firstly established for the vectorial Boolean negation \cite{guyeux10}. To obtain a characterization, we have introduced the notion of asynchronous iteration graph recalled bellow.
+
+Let $f$ be a map from $\mathds{B}^n$ to itself. The
+{\emph{asynchronous iteration graph}} associated with $f$ is the
+directed graph $\Gamma(f)$ defined by: the set of vertices is
+$\mathds{B}^n$; for all $x\in\mathds{B}^n$ and $i\in \llbracket1;n\rrbracket$,
+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 $x'$.
+
+We have finally proven in \cite{FCT11} that,
+
+
+\begin{theorem}
+\label{Th:Caractérisation des IC chaotiques}
+Let $f:\mathds{B}^n\to\mathds{B}^n$. $G_f$ is chaotic (according to Devaney)
+if and only if $\Gamma(f)$ is strongly connected.
+\end{theorem}
+
+
+
+
+\section{Application to Pseudo-Randomness}
+
+We have proposed in~\cite{bgw09:ip} a new family of generators that receives
+two PRNGs as inputs. These two generators are mixed with chaotic iterations,
+leading thus to a new PRNG that improves the statistical properties of each
+generator taken alone. Furthermore, our generator
+possesses various chaos properties
+that none of the generators used as input present.
+
+\begin{algorithm}[h!]
+%\begin{scriptsize}
+\KwIn{a function $f$, an iteration number $b$, an initial configuration $x^0$ ($n$ bits)}
+\KwOut{a configuration $x$ ($n$ bits)}
+$x\leftarrow x^0$\;
+$k\leftarrow b + \textit{XORshift}(b+1)$\;
+\For{$i=0,\dots,k-1$}
+{
+$s\leftarrow{\textit{XORshift}(n)}$\;
+$x\leftarrow{F_f(s,x)}$\;
+}
+return $x$\;
+%\end{scriptsize}
+\caption{PRNG with chaotic functions}
+\label{CI Algorithm}
+\end{algorithm}
+
+\begin{algorithm}[h!]
+\SetAlgoLined
+\KwIn{the internal configuration $z$ (a 32-bit word)}
+\KwOut{$y$ (a 32-bit word)}
+$z\leftarrow{z\oplus{(z\ll13)}}$\;
+$z\leftarrow{z\oplus{(z\gg17)}}$\;
+$z\leftarrow{z\oplus{(z\ll5)}}$\;
+$y\leftarrow{z}$\;
+return $y$\;
+\medskip
+\caption{An arbitrary round of \textit{XORshift} algorithm}
+\label{XORshift}
+\end{algorithm}
+
+
+
+
+
+This generator is synthesized in Algorithm~\ref{CI Algorithm}.
+It takes as input: a function $f$;
+an integer $b$, ensuring that the number of executed iterations is at least $b$ and at most $2b+1$; and an initial configuration $x^0$.
+It returns the new generated configuration $x$. Internally, it embeds two
+\textit{XORshift}$(k)$ PRNGs \cite{Marsaglia2003} that returns integers uniformly distributed
+into $\llbracket 1 ; k \rrbracket$.
+\textit{XORshift} is a category of very fast PRNGs designed by George Marsaglia, which repeatedly uses the transform of exclusive or (XOR, $\oplus$) on a number with a bit shifted version of it. This PRNG, which has a period of $2^{32}-1=4.29\times10^9$, is summed up in Algorithm~\ref{XORshift}. It is used in our PRNG to compute the strategy length and the strategy elements.
+
+
+We have proven in \cite{FCT11} that,
+
+\begin{theorem}
+ Let $f: \mathds{B}^{n} \rightarrow \mathds{B}^{n}$, $\Gamma(f)$ its
+ iteration graph, $\check{M}$ its adjacency
+ matrix and $M$ a $n\times n$ matrix defined as in the previous lemma.
+ If $\Gamma(f)$ is strongly connected, then
+ the output of the PRNG detailed in Algorithm~\ref{CI Algorithm} follows
+ a law that tends to the uniform distribution
+ if and only if $M$ is a double stochastic matrix.
+\end{theorem}
+
+
+\section{The relativity of disorder}
+\label{sec:de la relativité du désordre}
+
+\subsection{Impact of the topology's finenesse}
+
+Let us firstly introduce the following notations.
+
+\begin{notation}
+$\mathcal{X}_\tau$ will denote the topological space $\left(\mathcal{X},\tau\right)$, whereas $\mathcal{V}_\tau (x)$ will be the set of all the neighborhoods of $x$ when considering the topology $\tau$ (or simply $\mathcal{V} (x)$, if there is no ambiguity).
+\end{notation}
+
+
+
+\begin{theorem}
+\label{Th:chaos et finesse}
+Let $\mathcal{X}$ a set and $\tau, \tau'$ two topologies on $\mathcal{X}$ s.t. $\tau'$ is finer than $\tau$. Let $f:\mathcal{X} \to \mathcal{X}$, continuous both for $\tau$ and $\tau'$.
+
+If $(\mathcal{X}_{\tau'},f)$ is chaotic according to Devaney, then $(\mathcal{X}_\tau,f)$ is chaotic too.
+\end{theorem}
+
+\begin{proof}
+Let us firstly establish the transitivity of $(\mathcal{X}_\tau,f)$.
+
+Let $\omega_1, \omega_2$ two open sets of $\tau$. Then $\omega_1, \omega_2 \in \tau'$, becaus $\tau'$ is finer than $\tau$. As $f$ is $\tau'-$transitive, we can deduce that $\exists n \in \mathds{N}, \omega_1 \cap f^{(n)}(\omega_2) = \varnothing$. Consequently, $f$ is $\tau-$transitive.
+
+Let us now consider the regularity of $(\mathcal{X}_\tau,f)$, \emph{i.e.}, for all $x \in \mathcal{X}$, and for all $\tau-$neighborhood $V$ of $x$, there is a periodic point for $f$ into $V$.
+
+Let $x \in \mathcal{X}$ and $V \in \mathcal{V}_\tau (x)$ a $\tau-$neighborhood of $x$. By definition, $\exists \omega \in \tau, x \in \omega \subset V$.
+
+But $\tau \subset \tau'$, so $\omega \in \tau'$, and then $V \in \mathcal{V}_{\tau'} (x)$. As $(\mathcal{X}_{\tau'},f)$ is regular, there is a periodic point for $f$ into $V$, and the regularity of $(\mathcal{X}_\tau,f)$ is proven.
+\end{proof}
+
+\subsection{A given system can always be claimed as chaotic}
+
+Let $f$ an iteration function on $\mathcal{X}$ having at least a fixed point. Then this function is chaotic (in a certain way):
+
+\begin{theorem}
+Let $\mathcal{X}$ a nonempty set and $f: \mathcal{X} \to \X$ a function having at least a fixed point.
+Then $f$ is $\tau_0-$chaotic, where $\tau_0$ is the trivial (indiscrete) topology on $\X$.
+\end{theorem}
+
+
+\begin{proof}
+$f$ is transitive when $\forall \omega, \omega' \in \tau_0 \setminus \{\varnothing\}, \exists n \in \mathds{N}, f^{(n)}(\omega) \cap \omega' \neq \varnothing$.
+As $\tau_0 = \left\{ \varnothing, \X \right\}$, this is equivalent to look for an integer $n$ s.t. $f^{(n)}\left( \X \right) \cap \X \neq \varnothing$. For instance, $n=0$ is appropriate.
+
+Let us now consider $x \in \X$ and $V \in \mathcal{V}_{\tau_0} (x)$. Then $V = \mathcal{X}$, so $V$ has at least a fixed point for $f$. Consequently $f$ is regular, and the result is established.
+\end{proof}
+
+
+
+
+\subsection{A given system can always be claimed as non-chaotic}
+
+\begin{theorem}
+Let $\mathcal{X}$ be a set and $f: \mathcal{X} \to \X$.
+If $\X$ is infinite, then $\left( \X_{\tau_\infty}, f\right)$ is not chaotic (for the Devaney's formulation), where $\tau_\infty$ is the discrete topology.
+\end{theorem}
+
+\begin{proof}
+Let us prove it by contradiction, assuming that $\left(\X_{\tau_\infty}, f\right)$ is both transitive and regular.
+
+Let $x \in \X$ and $\{x\}$ one of its neighborhood. This neighborhood must contain a periodic point for $f$, if we want that $\left(\X_{\tau_\infty}, f\right)$ is regular. Then $x$ must be a periodic point of $f$.
+
+Let $I_x = \left\{ f^{(n)}(x), n \in \mathds{N}\right\}$. This set is finite because $x$ is periodic, and $\mathcal{X}$ is infinite, then $\exists y \in \mathcal{X}, y \notin I_x$.
+
+As $\left(\X_{\tau_\infty}, f\right)$ must be transitive, for all open nonempty sets $A$ and $B$, an integer $n$ must satisfy $f^{(n)}(A) \cap B \neq \varnothing$. However $\{x\}$ and $\{y\}$ are open sets and $y \notin I_x \Rightarrow \forall n, f^{(n)}\left( \{x\} \right) \cap \{y\} = \varnothing$.
+\end{proof}
+
+
+
+
-Présentation des itérations chaotiques
-\section{blabla}
+\section{Chaos on the order topology}
\subsection{The phase space is an interval of the real line}
\end{figure}
-We have written a Python program to represent the chaotic iterations with the vectorial negation on the real line $\mathds{R}$. Various representations of these CIs are given in Figures \ref{fig:ICs}, \ref{fig:ICs2} and \ref{fig:ICs3}. It can be remarked that the function $g$ is \alert{affine par morceaux}: it is linear on each interval having the form $\left[ \dfrac{n}{10}, \dfrac{n+1}{10}\right[$, $n \in \llbracket 0;2^{10}\times 10 \rrbracket$ \alert{and its line has a pent equal to 10}. Let us justify these claims:
+We have written a Python program to represent the chaotic iterations with the vectorial negation on the real line $\mathds{R}$. Various representations of these CIs are given in Figures \ref{fig:ICs}, \ref{fig:ICs2} and \ref{fig:ICs3}. It can be remarked that the function $g$ is a piecewise linear function: it is linear on each interval having the form $\left[ \dfrac{n}{10}, \dfrac{n+1}{10}\right[$, $n \in \llbracket 0;2^{10}\times 10 \rrbracket$ and its slope is equal to 10. Let us justify these claims:
\begin{proposition}
\label{Prop:derivabilite des ICs}
-Chaotic iterations $g$ defined on $\mathds{R}$ are \alert{infiniment dérivables} on $\big[ 0, 2^{10} \big[$, except on the 10241 points in $I$ defined by $\left\{ \dfrac{n}{10} ~\big/~ n \in \llbracket 0;2^{10}\times 10\rrbracket \right\}$.
+Chaotic iterations $g$ defined on $\mathds{R}$ have derivatives of all orders on $\big[ 0, 2^{10} \big[$, except on the 10241 points in $I$ defined by $\left\{ \dfrac{n}{10} ~\big/~ n \in \llbracket 0;2^{10}\times 10\rrbracket \right\}$.
-Furthermore, on each interval of the form $\left[ \dfrac{n}{10}, \dfrac{n+1}{10}\right[$, with $n \in \llbracket 0;2^{10}\times 10 \rrbracket$, the function $g$ is \emph{affine}. \alert{It is a line of pent equal to 10}: $\forall x \notin I, g'(x)=10$.
+Furthermore, on each interval of the form $\left[ \dfrac{n}{10}, \dfrac{n+1}{10}\right[$, with $n \in \llbracket 0;2^{10}\times 10 \rrbracket$, $g$ is a linear function, having a slope equal to 10: $\forall x \notin I, g'(x)=10$.
\end{proposition}
\end{proof}
\begin{remark}
-Finally, chaotic iterations are elements of the large family of functions that are \alert{chaotiques linéaires par morceaux}, like the tent map, the \alert{doublement de l'angle}, \emph{etc.}
+Finally, chaotic iterations are elements of the large family of functions that are both chaotic and piecewise linear (like the tent map).
\end{remark}
Additionally, $D_s(x^n, x) \to 0$, then $\forall k \in \mathds{N}^*, \exists N_k \in \mathds{N}, n \geqslant N_k \Rightarrow D_s(x^n,x) \leqslant 10^{-k}$. This means that for all $k$, an index $N_k$ can be found such that, $\forall n \geqslant N_k$, all the $x^n$ have the same $k$ firsts digits, which are the digits of $x$. We can deduce the convergence $\Delta(x^n,x) \to 0$, and thus the result.
\end{proof}
-The conclusion of these propositions is that the proposed metric is more \alert{précise} than the Euclidian distance, that is:
+The conclusion of these propositions is that the proposed metric is more precise than the Euclidian distance, that is:
\begin{corollary}
$D$ is finer than the Euclidian distance $\Delta$.
\subsubsection{Chaos according to Devaney}
-We have recalled previously that the chaotic iterations $\left(\Go, \mathcal{X}_d\right)$ are chaotic according to the formulation of Devaney. We can deduce that they are chaotic on $\mathds{R}$ too, when considering the \alert{topology of order}, because:
+We have recalled previously that the chaotic iterations $\left(\Go, \mathcal{X}_d\right)$ are chaotic according to the formulation of Devaney. We can deduce that they are chaotic on $\mathds{R}$ too, when considering the order topology, because:
\begin{itemize}
\item $\left(\Go, \mathcal{X}_d\right)$ and $\left(g, \big[ 0, 2^{10} \big[_D\right)$ are semiconjugate by $\varphi$,
\item Then $\left(g, \big[ 0, 2^{10} \big[_D\right)$ is a system chaotic according to Devaney, because the semiconjugacy preserve this character.
-\item But the topology generated by $D$ is finer than the topology generated by the Euclidian distance $\Delta$ -- which is the \alert{topology of order}.
-\item According to Theorem \ref{Th:chaos et finesse}, we can deduce that the chaotic iterations $g$ are indeed chaotic, as defined by Devaney, for the \alert{topology of order} on $\mathds{R}$.
+\item But the topology generated by $D$ is finer than the topology generated by the Euclidian distance $\Delta$ -- which is the order topology.
+\item According to Theorem \ref{Th:chaos et finesse}, we can deduce that the chaotic iterations $g$ are indeed chaotic, as defined by Devaney, for the order topology on $\mathds{R}$.
\end{itemize}
This result can be formulated as follows.
\begin{theorem}
\label{th:IC et topologie de l'ordre}
-The chaotic iterations $g$ on $\mathds{R}$ are chaotic according to the Devaney's formulation, when $\mathds{R}$ has his usual topology, which is the \alert{topology of order}.
+The chaotic iterations $g$ on $\mathds{R}$ are chaotic according to the Devaney's formulation, when $\mathds{R}$ has his usual topology, which is the order topology.
\end{theorem}
-Indeed this result is \alert{weaker} than the theorem establishing the chaos for the finer topology $d$. However the Theorem \ref{th:IC et topologie de l'ordre} still remains important. Indeed, we have studied in our previous works a set different from the usual set of study ($\mathcal{X}$ instead of $\mathds{R}$), in order to be as close as possible from the computer: the properties of disorder proved theoretically will then be preserved when computing. However, we could wonder whether this change does not lead to a disorder of a lower quality. In other words, have we replaced a situation of a good disorder lost when computing, to another situation of a disorder preserved but of bad quality. Theorem \ref{th:IC et topologie de l'ordre} prove exactly the contrary.
+Indeed this result is weaker than the theorem establishing the chaos for the finer topology $d$. However the Theorem \ref{th:IC et topologie de l'ordre} still remains important. Indeed, we have studied in our previous works a set different from the usual set of study ($\mathcal{X}$ instead of $\mathds{R}$), in order to be as close as possible from the computer: the properties of disorder proved theoretically will then be preserved when computing. However, we could wonder whether this change does not lead to a disorder of a lower quality. In other words, have we replaced a situation of a good disorder lost when computing, to another situation of a disorder preserved but of bad quality. Theorem \ref{th:IC et topologie de l'ordre} prove exactly the contrary.
