X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/rairo15.git/blobdiff_plain/44232560a3cd3ab96e3c35d7e4445f5185fb52d3..412f54fbed2c443f03aa09cc72f559221e63d83b:/intro.tex

diff --git a/intro.tex b/intro.tex
index b3e6754..3f51a08 100644
--- a/intro.tex
+++ b/intro.tex
@@ -16,14 +16,14 @@ a continuous function $k$ defined on a metrical space is said
 $x$ and all positive value $\epsilon$, it is possible to find another 
 point $y$, as close as possible to $x$, and an integer $t$ such that the distance
 between the $t$-th iterates of $x$ and $y$, denoted by $k^t(x)$ and $k^t(y)$,
-are larger than $\epsilon$. 
-% Cependant, dans sa dÃ©finition du chaos, 
-% Devaney~\cite{Devaney} impose Ã  la fonction chaotique deux autres propriÃ©tÃ©s 
-% appelÃ©es \emph{transitivitÃ©} et \emph{rÃ©gularitÃ©},
-% Les fonctions citÃ©es plus haut ont Ã©tÃ© Ã©tudiÃ©es 
-% au regard de ces propriÃ©tÃ©s et ont Ã©tÃ© prouvÃ©es comme chaotiques sur $\R$.
-% Cependant, rien ne garantit que ces propriÃ©tÃ©s sont prÃ©servÃ©es sur les nombres 
-% flottants qui est le domaine d'interprÃ©tation des nombres rÃ©els de $\R$.
+are larger than $\epsilon$. However, in his definition of chaos, Devaney~\cite{Devaney} 
+impose to the chaotic function two other properties called
+\emph{transitivity} and \emph{regularity}. Functions evoked above have
+been studied according to these properties, and they have been proven as chaotic on $\R$.
+But nothing guarantees that such properties are preserved when iterating the functions
+on floating point numbers, which is the domain of interpretation of real numbers $\R$ on
+machines.
+
 % 
 % Pour Ã©viter cette perte de chaos,  nous avons prÃ©sentÃ© des PRNGs qui itÃ¨rent des
 % fonctions continues  $G_f$ sur  un domaine discret  $\{ 1, \ldots,  n \}^{\Nats}
