-given region. More precisely,
-
-\begin{definition} \label{def2}
-A continuous function $f$ on a topological space $(\mathcal{X},\tau)$
-is defined to be {\bf topologically transitive} if for any pair of
-open sets $U$, $V \in \mathcal{X}$ there exists
-$k \in
-\mathds{N}^{\ast}$
- such that
-$f^k(U) \cap V \neq \emptyset$.
-\end{definition}
-
-This property implies that a dynamical system cannot be broken into
-simpler subsystems.
-Intuitively, its complexity does not allow any simplification.
-On the contrary, a dense set of periodic points is an
-element of regularity that a chaotic dynamical system has to exhibit.
-
-\begin{definition} \label{def3}
-A point $x$ is called a {\bf periodic point} for $f$ of period~$n \in
-\mathds{N}^{\ast}$ if $f^{n}(x)=x$.
-\end{definition}
-
-\begin{definition} \label{def4}
-$f$ is said to be {\bf regular} on $(\mathcal{X},\tau)$ if the set of
- periodic points for $f$ is dense in $\mathcal{X}$ ( for any $x \in
- \mathcal{X}$, we can find at least one periodic point in any of its
- neighborhood).
-\end{definition}
-
-This regularity ``counteracts'' the effects of transitivity. Thus,
-due to these two properties, two points close to each other can behave
-in a completely different manner, leading to unpredictability for the
-whole system. Then,
-
-\begin{definition} \label{sensitivity}
-$f$ has {\bf sensitive dependence on initial conditions} if there
- exists $\delta >0$ such that, for any $x\in \mathcal{X}$ and any
- neighborhood $V$ of $x$, there exist $y\in V$ and $n > 0$ such that
- $d\left(f^{n}(x), f^{n}(y)\right) >\delta $. The value $\delta$ is called the
- {\bf constant of sensitivity} of $f$.
-\end{definition}
-
-Finally,
-
-\begin{definition} \label{def5}
-$f$ is {\bf chaotic according to Devaney} on $(\mathcal{X},\tau)$ if
- $f$ is regular, topologically transitive, and has sensitive
- dependence to its initial conditions.
-\end{definition}
-
-Let us notice that for a metric space the last condition follows from
-the two first ones~\cite{Banks92}.
-
-\subsection{Chaotic Iterations}
-
-This section presents some basics on topological chaotic iterations.
+given region. This property implies that a dynamical system cannot be
+broken into simpler subsystems. Intuitively, its complexity does not
+allow any simplification.
+
+However, chaos needs some regularity to ``counteracts'' the effects of
+transitivity. % Thus, two points close to each other can behave in a completely different manner, the first one visiting the whole space whereas the second one has a regular orbit.
+In the Devaney's formulation, a dense set of periodic
+points is the element of regularity that a chaotic dynamical system has
+to exhibit.
+%\begin{definition} \label{def3}
+We recall that a point $x$ is a {\bf periodic point} for $f$ of
+period~$n \in \mathds{N}^{\ast}$ if $f^{n}(x)=x$.
+%\end{definition}
+Then, the map
+%\begin{definition} \label{def4}
+$f$ is {\bf regular} on the topological space $(\mathcal{X},\tau)$ if
+the set of periodic points for $f$ is dense in $\mathcal{X}$ (for any
+$x \in \mathcal{X}$, we can find at least one periodic point in any of
+its neighborhood).
+%\end{definition}
+Thus, due to these two properties, two points close to each other can
+behave in a completely different manner, leading to unpredictability
+for the whole system.
+
+Let us recall that $f$ has {\bf sensitive dependence on initial
+ conditions} if there exists $\delta >0$ such that, for any $x\in
+\mathcal{X}$ and any neighborhood $V$ of $x$, there exist $y\in V$ and
+$n > 0$ such that $d\left(f^{n}(x), f^{n}(y)\right) >\delta $. The
+value $\delta$ is called the {\bf constant of sensitivity} of $f$.
+
+Finally, the dynamical system that iterates $f$ is {\bf chaotic
+ according to Devaney} on $(\mathcal{X},\tau)$ if $f$ is regular,
+topologically transitive, and has sensitive dependence to its initial
+conditions. In what follows, iterations are said to be chaotic
+(according to Devaney) when the corresponding dynamical system is
+chaotic, as it is defined in the Devaney's formulation.
+
+%Let us notice that for a metric space the last condition follows from
+%the two first ones~\cite{Banks92}.
+
+\subsection{Asynchronous Iterations}
+
+%This section presents some basics on topological chaotic iterations.
