X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/review_prng.git/blobdiff_plain/73c00dda89e8836332b0c21c79bd398d3c87a05a..88947af28fe7e7f5b1dd35d21d71b2f69130b798:/review_prng.tex?ds=inline

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}{PropriÃ©tÃ©s de $(\mathcal{X},d)$}
-%  \begin{itemize}
-%    \item $\mathcal{X}$ est infini indÃ©nombrable
-%    \vspace{0.15cm}
-%    \item $(\mathcal{X},d)$ est un espace mÃ©trique 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 rÃ©versible.
-%  \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 pÃ©riodes}
-%% \begin{block}{MultiplicitÃ© des pÃ©riodes ?}
-%% Soit $f_0:\mathds{B}^\mathsf{N} \rightarrow \mathds{B}^\mathsf{N}$ la nÃ©gation 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 $G_{f_0}$ possÃ¨de plus de $n^2$ points pÃ©riodiques de pÃ©riode $2n$.
-%%     \end{itemize}
-%%   \end{itemize}
-%% \end{block}
-%% \uncover<2->{
-%%    Cette multiplicitÃ© des pÃ©riodes n'est pas le dÃ©sordre 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.
 
 
 
