From 2e8e9b41154df43b7a17bb19733abea069a7f20e Mon Sep 17 00:00:00 2001
From: guyeux <guyeux@gmail.com>
Date: Sun, 30 Oct 2011 11:04:56 +0100
Subject: [PATCH] Fin de la partie general chaootic iterations

---
 prng_gpu.tex | 43 +++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 43 insertions(+)

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 96098bb..5431409 100644
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -584,6 +584,49 @@ $G_{f}$ is consequently continuous.
 \end{proof}
 
 
+It is now possible to study the topological behavior of the general chaotic
+iterations. We will prove that,
+
+\begin{theorem}
+\label{t:chaos des general}
+ The general chaotic iterations defined on Equation~\ref{general CIs} satisfy
+the Devaney's property of chaos.
+\end{theorem}
+
+Let us firstly prove the following lemma.
+
+\begin{lemma}[Strong transitivity]
+\label{strongTrans}
+ For all couples $X,Y \in \mathcal{X}$ and any neighborhood $V$ of $X$, we can 
+find $n \in \mathds{N}^*$ and $X' \in V$ such that $G^n(X')=Y$.
+\end{lemma}
+
+\begin{proof}
+ Let $X=(S,E)$, $\varepsilon>0$, and $k_0 = \lfloor log_{10}(\varepsilon)+1 \rfloor$. 
+Any point $X'=(S',E')$ such that $E'=E$ and $\forall k \leqslant k_0, S'^k=S^k$, 
+are in the open ball $\mathcal{B}\left(X,\varepsilon\right)$. Let us define 
+$\check{X} = \left(\check{S},\check{E}\right)$, where $\check{X}= G^{k_0}(X)$.
+We denote by $s\subset \llbracket 1; \mathsf{N} \rrbracket$ the set of coordinates
+that are different between $\check{E}$ and the state of $Y$. Thus each point $X'$ of
+the form $(S',E')$ where $E'=E$ and $S'$ starts with 
+$(S^0, S^1, \hdots, S^{k_0},s,\hdots)$, verifies the following properties:
+\begin{itemize}
+ \item $X'$ is in $\mathcal{B}\left(X,\varepsilon\right)$,
+ \item the state of $G_f^{k_0+1}(X')$ is the state of $Y$.
+\end{itemize}
+Finally the point $\left(\left(S^0, S^1, \hdots, S^{k_0},s,s^0, s^1, \hdots\right); E\right)$, 
+where $(s^0,s^1, \hdots)$ is the strategy of $Y$, satisfies the properties
+claimed in the lemma.
+\end{proof}
+
+We can now prove the Theorem~\ref{t:chaos des general}...
+
+\begin{proof}[Theorem~\ref{t:chaos des general}]
+ On the one hand, strong transitivity implies transitivity. On the other hand, 
+the regularity is exactly Lemma~\ref{strongTrans} with $Y=X$. As the sensitivity
+to the initial condition is implied by these two properties, we thus have
+the theorem.
+\end{proof}
 
 
 
-- 
2.39.5