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},
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.
