From 88947af28fe7e7f5b1dd35d21d71b2f69130b798 Mon Sep 17 00:00:00 2001
From: guyeux <guyeux@gmail.com>
Date: Fri, 1 Jun 2012 14:33:47 +0200
Subject: [PATCH] Pouet

---
 mabase.bib      | 19 +++++++++++++++
 review_prng.tex | 61 ++++++++++++++++++++++---------------------------
 2 files changed, 46 insertions(+), 34 deletions(-)

diff --git a/mabase.bib b/mabase.bib
index 0033443..d5e366c 100644
--- a/mabase.bib
+++ b/mabase.bib
@@ -14,6 +14,25 @@
   timestamp = {2009.06.29}
 }
 
+
+@INCOLLECTION{gb11:bc,
+  author = {Guyeux, Christophe and Bahi, Jacques},
+  title = {A Topological Study of Chaotic Iterations. Application to Hash Functions},
+  booktitle = {CIPS, Computational Intelligence for Privacy and Security},
+  publisher = {Springer},
+  year = {2012},
+  volume = {394},
+  series = {Studies in Computational Intelligence},
+  pages = {51--73},
+  note = {Revised and extended journal version of an IJCNN best paper},
+  classement = {OS},
+  doi = {10.1007/978-3-642-25237-2_5},
+  domainehal = {INFO:INFO_DC, INFO:INFO_CR, INFO:INFO_MO},
+  equipe = {and},
+  inhal = {no},
+  url = {http://dx.doi.org/10.1007/978-3-642-25237-2_5}
+}
+
 @INPROCEEDINGS{BattiatoCGG99,
   author = {Sebastiano Battiato and Dario Catalano and Giovanni Gallo and Rosario
 	Gennaro},
diff --git a/review_prng.tex b/review_prng.tex
index aa5ebf2..0cdeb04 100644
--- a/review_prng.tex
+++ b/review_prng.tex
@@ -232,43 +232,36 @@ 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.
 
-% \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}
-
-% }
+\subsection{Topological Properties of Chaotic Iterations}
 
+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.
 
 
 
-- 
2.39.5