From: guyeux Date: Tue, 15 Nov 2011 15:27:10 +0000 (+0100) Subject: changement de la fin X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/commitdiff_plain/02b8e8e685923bf2b12d529361c3435c0291337d changement de la fin --- diff --git a/prng_gpu.tex b/prng_gpu.tex index 5b07118..150e434 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -978,530 +978,535 @@ In comparison, Listing~\ref{algo:seqCIprng} allows us to generate about -\section{The relativity of disorder} -\label{sec:de la relativité du désordre} +\section{Cryptanalysis of the Proposed PRNG} -In the next two sections, we investigate the impact of the choices that have -lead to the definitions of measures in Sections \ref{sec:chaotic iterations} and \ref{deuxième def}. -\subsection{Impact of the topology's finenesse} +Mettre ici la preuve de PCH -Let us firstly introduce the following notations. +%\section{The relativity of disorder} +%\label{sec:de la relativité du désordre} -\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} +%In the next two sections, we investigate the impact of the choices that have +%lead to the definitions of measures in Sections \ref{sec:chaotic iterations} and \ref{deuxième def}. +%\subsection{Impact of the topology's finenesse} +%Let us firstly introduce the following notations. -\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} - - +%\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} -\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. +%\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 $\tau'$. -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 $f$. +%If $(\mathcal{X}_{\tau'},f)$ is chaotic according to Devaney, then +%$(\mathcal{X}_\tau,f)$ is chaotic too. +%\end{theorem} -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 $\exists y \in -\mathcal{X}, y \notin I_x$. +%\begin{proof} +%Let us firstly establish the transitivity of $(\mathcal{X}_\tau,f)$. -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} +%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): -\section{Chaos on the order topology} -\label{sec: chaos order topology} -\subsection{The phase space is an interval of the real line} +%\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} -\subsubsection{Toward a topological semiconjugacy} -In what follows, our intention is to establish, by using a topological -semiconjugacy, that chaotic iterations over $\mathcal{X}$ can be described as -iterations on a real interval. To do so, we must firstly introduce some -notations and terminologies. +%\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 $\mathcal{S}_\mathsf{N}$ be the set of sequences belonging into $\llbracket -1; \mathsf{N}\rrbracket$ and $\mathcal{X}_{\mathsf{N}} = \mathcal{S}_\mathsf{N} -\times \B^\mathsf{N}$. +%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} -\begin{definition} -The function $\varphi: \mathcal{S}_{10} \times\mathds{B}^{10} \rightarrow \big[ -0, 2^{10} \big[$ is defined by: -\begin{equation} - \begin{array}{cccl} -\varphi: & \mathcal{X}_{10} = \mathcal{S}_{10} \times\mathds{B}^{10}& -\longrightarrow & \big[ 0, 2^{10} \big[ \\ - & (S,E) = \left((S^0, S^1, \hdots ); (E_0, \hdots, E_9)\right) & \longmapsto & -\varphi \left((S,E)\right) -\end{array} -\end{equation} -where $\varphi\left((S,E)\right)$ is the real number: -\begin{itemize} -\item whose integral part $e$ is $\displaystyle{\sum_{k=0}^9 2^{9-k} E_k}$, that -is, the binary digits of $e$ are $E_0 ~ E_1 ~ \hdots ~ E_9$. -\item whose decimal part $s$ is equal to $s = 0,S^0~ S^1~ S^2~ \hdots = -\sum_{k=1}^{+\infty} 10^{-k} S^{k-1}.$ -\end{itemize} -\end{definition} +%\subsection{A given system can always be claimed as non-chaotic} -$\varphi$ realizes the association between a point of $\mathcal{X}_{10}$ and a -real number into $\big[ 0, 2^{10} \big[$. We must now translate the chaotic -iterations $\Go$ on this real interval. To do so, two intermediate functions -over $\big[ 0, 2^{10} \big[$ must be introduced: +%\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. -\begin{definition} -\label{def:e et s} -Let $x \in \big[ 0, 2^{10} \big[$ and: -\begin{itemize} -\item $e_0, \hdots, e_9$ the binary digits of the integral part of $x$: -$\displaystyle{\lfloor x \rfloor = \sum_{k=0}^{9} 2^{9-k} e_k}$. -\item $(s^k)_{k\in \mathds{N}}$ the digits of $x$, where the chosen decimal -decomposition of $x$ is the one that does not have an infinite number of 9: -$\displaystyle{x = \lfloor x \rfloor + \sum_{k=0}^{+\infty} s^k 10^{-k-1}}$. -\end{itemize} -$e$ and $s$ are thus defined as follows: -\begin{equation} -\begin{array}{cccl} -e: & \big[ 0, 2^{10} \big[ & \longrightarrow & \mathds{B}^{10} \\ - & x & \longmapsto & (e_0, \hdots, e_9) -\end{array} -\end{equation} -and -\begin{equation} - \begin{array}{cccc} -s: & \big[ 0, 2^{10} \big[ & \longrightarrow & \llbracket 0, 9 -\rrbracket^{\mathds{N}} \\ - & x & \longmapsto & (s^k)_{k \in \mathds{N}} -\end{array} -\end{equation} -\end{definition} - -We are now able to define the function $g$, whose goal is to translate the -chaotic iterations $\Go$ on an interval of $\mathds{R}$. +%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} + + + + + + +%\section{Chaos on the order topology} +%\label{sec: chaos order topology} +%\subsection{The phase space is an interval of the real line} + +%\subsubsection{Toward a topological semiconjugacy} + +%In what follows, our intention is to establish, by using a topological +%semiconjugacy, that chaotic iterations over $\mathcal{X}$ can be described as +%iterations on a real interval. To do so, we must firstly introduce some +%notations and terminologies. + +%Let $\mathcal{S}_\mathsf{N}$ be the set of sequences belonging into $\llbracket +%1; \mathsf{N}\rrbracket$ and $\mathcal{X}_{\mathsf{N}} = \mathcal{S}_\mathsf{N} +%\times \B^\mathsf{N}$. + + +%\begin{definition} +%The function $\varphi: \mathcal{S}_{10} \times\mathds{B}^{10} \rightarrow \big[ +%0, 2^{10} \big[$ is defined by: +%\begin{equation} +% \begin{array}{cccl} +%\varphi: & \mathcal{X}_{10} = \mathcal{S}_{10} \times\mathds{B}^{10}& +%\longrightarrow & \big[ 0, 2^{10} \big[ \\ +% & (S,E) = \left((S^0, S^1, \hdots ); (E_0, \hdots, E_9)\right) & \longmapsto & +%\varphi \left((S,E)\right) +%\end{array} +%\end{equation} +%where $\varphi\left((S,E)\right)$ is the real number: +%\begin{itemize} +%\item whose integral part $e$ is $\displaystyle{\sum_{k=0}^9 2^{9-k} E_k}$, that +%is, the binary digits of $e$ are $E_0 ~ E_1 ~ \hdots ~ E_9$. +%\item whose decimal part $s$ is equal to $s = 0,S^0~ S^1~ S^2~ \hdots = +%\sum_{k=1}^{+\infty} 10^{-k} S^{k-1}.$ +%\end{itemize} +%\end{definition} + + + +%$\varphi$ realizes the association between a point of $\mathcal{X}_{10}$ and a +%real number into $\big[ 0, 2^{10} \big[$. We must now translate the chaotic +%iterations $\Go$ on this real interval. To do so, two intermediate functions +%over $\big[ 0, 2^{10} \big[$ must be introduced: + + +%\begin{definition} +%\label{def:e et s} +%Let $x \in \big[ 0, 2^{10} \big[$ and: +%\begin{itemize} +%\item $e_0, \hdots, e_9$ the binary digits of the integral part of $x$: +%$\displaystyle{\lfloor x \rfloor = \sum_{k=0}^{9} 2^{9-k} e_k}$. +%\item $(s^k)_{k\in \mathds{N}}$ the digits of $x$, where the chosen decimal +%decomposition of $x$ is the one that does not have an infinite number of 9: +%$\displaystyle{x = \lfloor x \rfloor + \sum_{k=0}^{+\infty} s^k 10^{-k-1}}$. +%\end{itemize} +%$e$ and $s$ are thus defined as follows: +%\begin{equation} +%\begin{array}{cccl} +%e: & \big[ 0, 2^{10} \big[ & \longrightarrow & \mathds{B}^{10} \\ +% & x & \longmapsto & (e_0, \hdots, e_9) +%\end{array} +%\end{equation} +%and +%\begin{equation} +% \begin{array}{cccc} +%s: & \big[ 0, 2^{10} \big[ & \longrightarrow & \llbracket 0, 9 +%\rrbracket^{\mathds{N}} \\ +% & x & \longmapsto & (s^k)_{k \in \mathds{N}} +%\end{array} +%\end{equation} +%\end{definition} + +%We are now able to define the function $g$, whose goal is to translate the +%chaotic iterations $\Go$ on an interval of $\mathds{R}$. + +%\begin{definition} +%$g:\big[ 0, 2^{10} \big[ \longrightarrow \big[ 0, 2^{10} \big[$ is defined by: +%\begin{equation} +%\begin{array}{cccc} +%g: & \big[ 0, 2^{10} \big[ & \longrightarrow & \big[ 0, 2^{10} \big[ \\ +% & x & \longmapsto & g(x) +%\end{array} +%\end{equation} +%where g(x) is the real number of $\big[ 0, 2^{10} \big[$ defined bellow: +%\begin{itemize} +%\item its integral part has a binary decomposition equal to $e_0', \hdots, +%e_9'$, with: +% \begin{equation} +%e_i' = \left\{ +%\begin{array}{ll} +%e(x)_i & \textrm{ if } i \neq s^0\\ +%e(x)_i + 1 \textrm{ (mod 2)} & \textrm{ if } i = s^0\\ +%\end{array} +%\right. +%\end{equation} +%\item whose decimal part is $s(x)^1, s(x)^2, \hdots$ +%\end{itemize} +%\end{definition} + +%\bigskip + + +%In other words, if $x = \displaystyle{\sum_{k=0}^{9} 2^{9-k} e_k + +%\sum_{k=0}^{+\infty} s^{k} ~10^{-k-1}}$, then: +%\begin{equation} +%g(x) = +%\displaystyle{\sum_{k=0}^{9} 2^{9-k} (e_k + \delta(k,s^0) \textrm{ (mod 2)}) + +%\sum_{k=0}^{+\infty} s^{k+1} 10^{-k-1}}. +%\end{equation} + + +%\subsubsection{Defining a metric on $\big[ 0, 2^{10} \big[$} + +%Numerous metrics can be defined on the set $\big[ 0, 2^{10} \big[$, the most +%usual one being the Euclidian distance recalled bellow: + +%\begin{notation} +%\index{distance!euclidienne} +%$\Delta$ is the Euclidian distance on $\big[ 0, 2^{10} \big[$, that is, +%$\Delta(x,y) = |y-x|^2$. +%\end{notation} + +%\medskip + +%This Euclidian distance does not reproduce exactly the notion of proximity +%induced by our first distance $d$ on $\X$. Indeed $d$ is finer than $\Delta$. +%This is the reason why we have to introduce the following metric: + + + +%\begin{definition} +%Let $x,y \in \big[ 0, 2^{10} \big[$. +%$D$ denotes the function from $\big[ 0, 2^{10} \big[^2$ to $\mathds{R}^+$ +%defined by: $D(x,y) = D_e\left(e(x),e(y)\right) + D_s\left(s(x),s(y)\right)$, +%where: +%\begin{center} +%$\displaystyle{D_e(E,\check{E}) = \sum_{k=0}^\mathsf{9} \delta (E_k, +%\check{E}_k)}$, ~~and~ $\displaystyle{D_s(S,\check{S}) = \sum_{k = 1}^\infty +%\dfrac{|S^k-\check{S}^k|}{10^k}}$. +%\end{center} +%\end{definition} + +%\begin{proposition} +%$D$ is a distance on $\big[ 0, 2^{10} \big[$. +%\end{proposition} + +%\begin{proof} +%The three axioms defining a distance must be checked. +%\begin{itemize} +%\item $D \geqslant 0$, because everything is positive in its definition. If +%$D(x,y)=0$, then $D_e(x,y)=0$, so the integral parts of $x$ and $y$ are equal +%(they have the same binary decomposition). Additionally, $D_s(x,y) = 0$, then +%$\forall k \in \mathds{N}^*, s(x)^k = s(y)^k$. In other words, $x$ and $y$ have +%the same $k-$th decimal digit, $\forall k \in \mathds{N}^*$. And so $x=y$. +%\item $D(x,y)=D(y,x)$. +%\item Finally, the triangular inequality is obtained due to the fact that both +%$\delta$ and $\Delta(x,y)=|x-y|$ satisfy it. +%\end{itemize} +%\end{proof} + + +%The convergence of sequences according to $D$ is not the same than the usual +%convergence related to the Euclidian metric. For instance, if $x^n \to x$ +%according to $D$, then necessarily the integral part of each $x^n$ is equal to +%the integral part of $x$ (at least after a given threshold), and the decimal +%part of $x^n$ corresponds to the one of $x$ ``as far as required''. +%To illustrate this fact, a comparison between $D$ and the Euclidian distance is +%given Figure \ref{fig:comparaison de distances}. These illustrations show that +%$D$ is richer and more refined than the Euclidian distance, and thus is more +%precise. + + +%\begin{figure}[t] +%\begin{center} +% \subfigure[Function $x \to dist(x;1,234) $ on the interval +%$(0;5)$.]{\includegraphics[scale=.35]{DvsEuclidien.pdf}}\quad +% \subfigure[Function $x \to dist(x;3) $ on the interval +%$(0;5)$.]{\includegraphics[scale=.35]{DvsEuclidien2.pdf}} +%\end{center} +%\caption{Comparison between $D$ (in blue) and the Euclidian distane (in green).} +%\label{fig:comparaison de distances} +%\end{figure} + + + + +%\subsubsection{The semiconjugacy} + +%It is now possible to define a topological semiconjugacy between $\mathcal{X}$ +%and an interval of $\mathds{R}$: + +%\begin{theorem} +%Chaotic iterations on the phase space $\mathcal{X}$ are simple iterations on +%$\mathds{R}$, which is illustrated by the semiconjugacy of the diagram bellow: +%\begin{equation*} +%\begin{CD} +%\left(~\mathcal{S}_{10} \times\mathds{B}^{10}, d~\right) @>G_{f_0}>> +%\left(~\mathcal{S}_{10} \times\mathds{B}^{10}, d~\right)\\ +% @V{\varphi}VV @VV{\varphi}V\\ +%\left( ~\big[ 0, 2^{10} \big[, D~\right) @>>g> \left(~\big[ 0, 2^{10} \big[, +%D~\right) +%\end{CD} +%\end{equation*} +%\end{theorem} + +%\begin{proof} +%$\varphi$ has been constructed in order to be continuous and onto. +%\end{proof} + +%In other words, $\mathcal{X}$ is approximately equal to $\big[ 0, 2^\mathsf{N} +%\big[$. + + + + + + +%\subsection{Study of the chaotic iterations described as a real function} + + +%\begin{figure}[t] +%\begin{center} +% \subfigure[ICs on the interval +%$(0,9;1)$.]{\includegraphics[scale=.35]{ICs09a1.pdf}}\quad +% \subfigure[ICs on the interval +%$(0,7;1)$.]{\includegraphics[scale=.35]{ICs07a95.pdf}}\\ +% \subfigure[ICs on the interval +%$(0,5;1)$.]{\includegraphics[scale=.35]{ICs05a1.pdf}}\quad +% \subfigure[ICs on the interval +%$(0;1)$]{\includegraphics[scale=.35]{ICs0a1.pdf}} +%\end{center} +%\caption{Representation of the chaotic iterations.} +%\label{fig:ICs} +%\end{figure} + + + + +%\begin{figure}[t] +%\begin{center} +% \subfigure[ICs on the interval +%$(510;514)$.]{\includegraphics[scale=.35]{ICs510a514.pdf}}\quad +% \subfigure[ICs on the interval +%$(1000;1008)$]{\includegraphics[scale=.35]{ICs1000a1008.pdf}} +%\end{center} +%\caption{ICs on small intervals.} +%\label{fig:ICs2} +%\end{figure} + +%\begin{figure}[t] +%\begin{center} +% \subfigure[ICs on the interval +%$(0;16)$.]{\includegraphics[scale=.3]{ICs0a16.pdf}}\quad +% \subfigure[ICs on the interval +%$(40;70)$.]{\includegraphics[scale=.45]{ICs40a70.pdf}}\quad +%\end{center} +%\caption{General aspect of the chaotic iterations.} +%\label{fig:ICs3} +%\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 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 $\mathds{R}$ 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 $n \in \llbracket 0;2^{10}\times 10 \rrbracket$, +%$g$ is a linear function, having a slope equal to 10: $\forall x \notin I, +%g'(x)=10$. +%\end{proposition} -\begin{definition} -$g:\big[ 0, 2^{10} \big[ \longrightarrow \big[ 0, 2^{10} \big[$ is defined by: -\begin{equation} -\begin{array}{cccc} -g: & \big[ 0, 2^{10} \big[ & \longrightarrow & \big[ 0, 2^{10} \big[ \\ - & x & \longmapsto & g(x) -\end{array} -\end{equation} -where g(x) is the real number of $\big[ 0, 2^{10} \big[$ defined bellow: -\begin{itemize} -\item its integral part has a binary decomposition equal to $e_0', \hdots, -e_9'$, with: - \begin{equation} -e_i' = \left\{ -\begin{array}{ll} -e(x)_i & \textrm{ if } i \neq s^0\\ -e(x)_i + 1 \textrm{ (mod 2)} & \textrm{ if } i = s^0\\ -\end{array} -\right. -\end{equation} -\item whose decimal part is $s(x)^1, s(x)^2, \hdots$ -\end{itemize} -\end{definition} - -\bigskip - - -In other words, if $x = \displaystyle{\sum_{k=0}^{9} 2^{9-k} e_k + -\sum_{k=0}^{+\infty} s^{k} ~10^{-k-1}}$, then: -\begin{equation} -g(x) = -\displaystyle{\sum_{k=0}^{9} 2^{9-k} (e_k + \delta(k,s^0) \textrm{ (mod 2)}) + -\sum_{k=0}^{+\infty} s^{k+1} 10^{-k-1}}. -\end{equation} - - -\subsubsection{Defining a metric on $\big[ 0, 2^{10} \big[$} - -Numerous metrics can be defined on the set $\big[ 0, 2^{10} \big[$, the most -usual one being the Euclidian distance recalled bellow: - -\begin{notation} -\index{distance!euclidienne} -$\Delta$ is the Euclidian distance on $\big[ 0, 2^{10} \big[$, that is, -$\Delta(x,y) = |y-x|^2$. -\end{notation} - -\medskip - -This Euclidian distance does not reproduce exactly the notion of proximity -induced by our first distance $d$ on $\X$. Indeed $d$ is finer than $\Delta$. -This is the reason why we have to introduce the following metric: - - - -\begin{definition} -Let $x,y \in \big[ 0, 2^{10} \big[$. -$D$ denotes the function from $\big[ 0, 2^{10} \big[^2$ to $\mathds{R}^+$ -defined by: $D(x,y) = D_e\left(e(x),e(y)\right) + D_s\left(s(x),s(y)\right)$, -where: -\begin{center} -$\displaystyle{D_e(E,\check{E}) = \sum_{k=0}^\mathsf{9} \delta (E_k, -\check{E}_k)}$, ~~and~ $\displaystyle{D_s(S,\check{S}) = \sum_{k = 1}^\infty -\dfrac{|S^k-\check{S}^k|}{10^k}}$. -\end{center} -\end{definition} - -\begin{proposition} -$D$ is a distance on $\big[ 0, 2^{10} \big[$. -\end{proposition} - -\begin{proof} -The three axioms defining a distance must be checked. -\begin{itemize} -\item $D \geqslant 0$, because everything is positive in its definition. If -$D(x,y)=0$, then $D_e(x,y)=0$, so the integral parts of $x$ and $y$ are equal -(they have the same binary decomposition). Additionally, $D_s(x,y) = 0$, then -$\forall k \in \mathds{N}^*, s(x)^k = s(y)^k$. In other words, $x$ and $y$ have -the same $k-$th decimal digit, $\forall k \in \mathds{N}^*$. And so $x=y$. -\item $D(x,y)=D(y,x)$. -\item Finally, the triangular inequality is obtained due to the fact that both -$\delta$ and $\Delta(x,y)=|x-y|$ satisfy it. -\end{itemize} -\end{proof} - - -The convergence of sequences according to $D$ is not the same than the usual -convergence related to the Euclidian metric. For instance, if $x^n \to x$ -according to $D$, then necessarily the integral part of each $x^n$ is equal to -the integral part of $x$ (at least after a given threshold), and the decimal -part of $x^n$ corresponds to the one of $x$ ``as far as required''. -To illustrate this fact, a comparison between $D$ and the Euclidian distance is -given Figure \ref{fig:comparaison de distances}. These illustrations show that -$D$ is richer and more refined than the Euclidian distance, and thus is more -precise. - - -\begin{figure}[t] -\begin{center} - \subfigure[Function $x \to dist(x;1,234) $ on the interval -$(0;5)$.]{\includegraphics[scale=.35]{DvsEuclidien.pdf}}\quad - \subfigure[Function $x \to dist(x;3) $ on the interval -$(0;5)$.]{\includegraphics[scale=.35]{DvsEuclidien2.pdf}} -\end{center} -\caption{Comparison between $D$ (in blue) and the Euclidian distane (in green).} -\label{fig:comparaison de distances} -\end{figure} - - - - -\subsubsection{The semiconjugacy} - -It is now possible to define a topological semiconjugacy between $\mathcal{X}$ -and an interval of $\mathds{R}$: - -\begin{theorem} -Chaotic iterations on the phase space $\mathcal{X}$ are simple iterations on -$\mathds{R}$, which is illustrated by the semiconjugacy of the diagram bellow: -\begin{equation*} -\begin{CD} -\left(~\mathcal{S}_{10} \times\mathds{B}^{10}, d~\right) @>G_{f_0}>> -\left(~\mathcal{S}_{10} \times\mathds{B}^{10}, d~\right)\\ - @V{\varphi}VV @VV{\varphi}V\\ -\left( ~\big[ 0, 2^{10} \big[, D~\right) @>>g> \left(~\big[ 0, 2^{10} \big[, -D~\right) -\end{CD} -\end{equation*} -\end{theorem} - -\begin{proof} -$\varphi$ has been constructed in order to be continuous and onto. -\end{proof} - -In other words, $\mathcal{X}$ is approximately equal to $\big[ 0, 2^\mathsf{N} -\big[$. - - - - - - -\subsection{Study of the chaotic iterations described as a real function} - - -\begin{figure}[t] -\begin{center} - \subfigure[ICs on the interval -$(0,9;1)$.]{\includegraphics[scale=.35]{ICs09a1.pdf}}\quad - \subfigure[ICs on the interval -$(0,7;1)$.]{\includegraphics[scale=.35]{ICs07a95.pdf}}\\ - \subfigure[ICs on the interval -$(0,5;1)$.]{\includegraphics[scale=.35]{ICs05a1.pdf}}\quad - \subfigure[ICs on the interval -$(0;1)$]{\includegraphics[scale=.35]{ICs0a1.pdf}} -\end{center} -\caption{Representation of the chaotic iterations.} -\label{fig:ICs} -\end{figure} - - - - -\begin{figure}[t] -\begin{center} - \subfigure[ICs on the interval -$(510;514)$.]{\includegraphics[scale=.35]{ICs510a514.pdf}}\quad - \subfigure[ICs on the interval -$(1000;1008)$]{\includegraphics[scale=.35]{ICs1000a1008.pdf}} -\end{center} -\caption{ICs on small intervals.} -\label{fig:ICs2} -\end{figure} - -\begin{figure}[t] -\begin{center} - \subfigure[ICs on the interval -$(0;16)$.]{\includegraphics[scale=.3]{ICs0a16.pdf}}\quad - \subfigure[ICs on the interval -$(40;70)$.]{\includegraphics[scale=.45]{ICs40a70.pdf}}\quad -\end{center} -\caption{General aspect of the chaotic iterations.} -\label{fig:ICs3} -\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 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 $\mathds{R}$ 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 $n \in \llbracket 0;2^{10}\times 10 \rrbracket$, -$g$ is a linear function, having a slope equal to 10: $\forall x \notin I, -g'(x)=10$. -\end{proposition} - - -\begin{proof} -Let $I_n = \left[ \dfrac{n}{10}, \dfrac{n+1}{10}\right[$, with $n \in \llbracket -0;2^{10}\times 10 \rrbracket$. All the points of $I_n$ have the same integral -prat $e$ and the same decimal part $s^0$: on the set $I_n$, functions $e(x)$ -and $x \mapsto s(x)^0$ of Definition \ref{def:e et s} only depend on $n$. So all -the images $g(x)$ of these points $x$: -\begin{itemize} -\item Have the same integral part, which is $e$, except probably the bit number -$s^0$. In other words, this integer has approximately the same binary -decomposition than $e$, the sole exception being the digit $s^0$ (this number is -then either $e+2^{10-s^0}$ or $e-2^{10-s^0}$, depending on the parity of $s^0$, -\emph{i.e.}, it is equal to $e+(-1)^{s^0}\times 2^{10-s^0}$). -\item A shift to the left has been applied to the decimal part $y$, losing by -doing so the common first digit $s^0$. In other words, $y$ has been mapped into -$10\times y - s^0$. -\end{itemize} -To sum up, the action of $g$ on the points of $I$ is as follows: first, make a -multiplication by 10, and second, add the same constant to each term, which is -$\dfrac{1}{10}\left(e+(-1)^{s^0}\times 2^{10-s^0}\right)-s^0$. -\end{proof} - -\begin{remark} -Finally, chaotic iterations are elements of the large family of functions that -are both chaotic and piecewise linear (like the tent map). -\end{remark} - - - -\subsection{Comparison of the two metrics on $\big[ 0, 2^\mathsf{N} \big[$} - -The two propositions bellow allow to compare our two distances on $\big[ 0, -2^\mathsf{N} \big[$: - -\begin{proposition} -Id: $\left(~\big[ 0, 2^\mathsf{N} \big[,\Delta~\right) \to \left(~\big[ 0, -2^\mathsf{N} \big[, D~\right)$ is not continuous. -\end{proposition} - -\begin{proof} -The sequence $x^n = 1,999\hdots 999$ constituted by $n$ 9 as decimal part, is -such that: -\begin{itemize} -\item $\Delta (x^n,2) \to 0.$ -\item But $D(x^n,2) \geqslant 1$, then $D(x^n,2)$ does not converge to 0. -\end{itemize} - -The sequential characterization of the continuity concludes the demonstration. -\end{proof} - - - -A contrario: - -\begin{proposition} -Id: $\left(~\big[ 0, 2^\mathsf{N} \big[,D~\right) \to \left(~\big[ 0, -2^\mathsf{N} \big[, \Delta ~\right)$ is a continuous fonction. -\end{proposition} - -\begin{proof} -If $D(x^n,x) \to 0$, then $D_e(x^n,x) = 0$ at least for $n$ larger than a given -threshold, because $D_e$ only returns integers. So, after this threshold, the -integral parts of all the $x^n$ are equal to the integral part of $x$. - -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 precise -than the Euclidian distance, that is: - -\begin{corollary} -$D$ is finer than the Euclidian distance $\Delta$. -\end{corollary} - -This corollary can be reformulated as follows: - -\begin{itemize} -\item The topology produced by $\Delta$ is a subset of the topology produced by -$D$. -\item $D$ has more open sets than $\Delta$. -\item It is harder to converge for the topology $\tau_D$ inherited by $D$, than -to converge with the one inherited by $\Delta$, which is denoted here by -$\tau_\Delta$. -\end{itemize} - - -\subsection{Chaos of the chaotic iterations on $\mathds{R}$} -\label{chpt:Chaos des itérations chaotiques sur R} - - - -\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 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 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 -order topology. -\end{theorem} -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. - +%\begin{proof} +%Let $I_n = \left[ \dfrac{n}{10}, \dfrac{n+1}{10}\right[$, with $n \in \llbracket +%0;2^{10}\times 10 \rrbracket$. All the points of $I_n$ have the same integral +%prat $e$ and the same decimal part $s^0$: on the set $I_n$, functions $e(x)$ +%and $x \mapsto s(x)^0$ of Definition \ref{def:e et s} only depend on $n$. So all +%the images $g(x)$ of these points $x$: +%\begin{itemize} +%\item Have the same integral part, which is $e$, except probably the bit number +%$s^0$. In other words, this integer has approximately the same binary +%decomposition than $e$, the sole exception being the digit $s^0$ (this number is +%then either $e+2^{10-s^0}$ or $e-2^{10-s^0}$, depending on the parity of $s^0$, +%\emph{i.e.}, it is equal to $e+(-1)^{s^0}\times 2^{10-s^0}$). +%\item A shift to the left has been applied to the decimal part $y$, losing by +%doing so the common first digit $s^0$. In other words, $y$ has been mapped into +%$10\times y - s^0$. +%\end{itemize} +%To sum up, the action of $g$ on the points of $I$ is as follows: first, make a +%multiplication by 10, and second, add the same constant to each term, which is +%$\dfrac{1}{10}\left(e+(-1)^{s^0}\times 2^{10-s^0}\right)-s^0$. +%\end{proof} + +%\begin{remark} +%Finally, chaotic iterations are elements of the large family of functions that +%are both chaotic and piecewise linear (like the tent map). +%\end{remark} + + + +%\subsection{Comparison of the two metrics on $\big[ 0, 2^\mathsf{N} \big[$} + +%The two propositions bellow allow to compare our two distances on $\big[ 0, +%2^\mathsf{N} \big[$: + +%\begin{proposition} +%Id: $\left(~\big[ 0, 2^\mathsf{N} \big[,\Delta~\right) \to \left(~\big[ 0, +%2^\mathsf{N} \big[, D~\right)$ is not continuous. +%\end{proposition} + +%\begin{proof} +%The sequence $x^n = 1,999\hdots 999$ constituted by $n$ 9 as decimal part, is +%such that: +%\begin{itemize} +%\item $\Delta (x^n,2) \to 0.$ +%\item But $D(x^n,2) \geqslant 1$, then $D(x^n,2)$ does not converge to 0. +%\end{itemize} + +%The sequential characterization of the continuity concludes the demonstration. +%\end{proof} + + + +%A contrario: + +%\begin{proposition} +%Id: $\left(~\big[ 0, 2^\mathsf{N} \big[,D~\right) \to \left(~\big[ 0, +%2^\mathsf{N} \big[, \Delta ~\right)$ is a continuous fonction. +%\end{proposition} + +%\begin{proof} +%If $D(x^n,x) \to 0$, then $D_e(x^n,x) = 0$ at least for $n$ larger than a given +%threshold, because $D_e$ only returns integers. So, after this threshold, the +%integral parts of all the $x^n$ are equal to the integral part of $x$. + +%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 precise +%than the Euclidian distance, that is: + +%\begin{corollary} +%$D$ is finer than the Euclidian distance $\Delta$. +%\end{corollary} + +%This corollary can be reformulated as follows: + +%\begin{itemize} +%\item The topology produced by $\Delta$ is a subset of the topology produced by +%$D$. +%\item $D$ has more open sets than $\Delta$. +%\item It is harder to converge for the topology $\tau_D$ inherited by $D$, than +%to converge with the one inherited by $\Delta$, which is denoted here by +%$\tau_\Delta$. +%\end{itemize} + + +%\subsection{Chaos of the chaotic iterations on $\mathds{R}$} +%\label{chpt:Chaos des itérations chaotiques sur R} + + + +%\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 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 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 +%order topology. +%\end{theorem} + +%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. +%