From 3add2868094bc67744e89777e13ce7087b53e01e Mon Sep 17 00:00:00 2001 From: guyeux Date: Sun, 30 Oct 2011 17:02:33 +0100 Subject: [PATCH] Correction d'une preuve --- prng_gpu.tex | 25 +++++++++++++++++++++---- 1 file changed, 21 insertions(+), 4 deletions(-) diff --git a/prng_gpu.tex b/prng_gpu.tex index 11dd246..c504e98 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -623,10 +623,27 @@ claimed in the lemma. 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. +Firstly, strong transitivity implies transitivity. + +Let $(S,E) \in\mathcal{X}$ and $\varepsilon >0$. To +prove that $G_f$ is regular, it is sufficient to prove that +there exists a strategy $\tilde S$ such that the distance between +$(\tilde S,E)$ and $(S,E)$ is less than $\varepsilon$, and such that +$(\tilde S,E)$ is a periodic point. + +Let $t_1=\lfloor-\log_{10}(\varepsilon)\rfloor$, and let $E'$ be the +configuration that we obtain from $(S,E)$ after $t_1$ iterations of +$G_f$. As $G_f$ is strongly transitive, there exists a strategy $S'$ +and $t_2\in\mathds{N}$ such +that $E$ is reached from $(S',E')$ after $t_2$ iterations of $G_f$. + +Consider the strategy $\tilde S$ that alternates the first $t_1$ terms +of $S$ and the first $t_2$ terms of $S'$: $$\tilde +S=(S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots).$$ It +is clear that $(\tilde S,E)$ is obtained from $(\tilde S,E)$ after +$t_1+t_2$ iterations of $G_f$. So $(\tilde S,E)$ is a periodic +point. Since $\tilde S_t=S_t$ for $t