\begin{block}{Definition: Chaotic function [4]$^4$}
Let $(\mathcal{X}; d)$ be a metric space.
A function $f: \mathcal{X} \rightarrow \mathcal{X}$ is chaotic on $\mathcal{X}$ if:
\begin{enumerate}
\item $f$: topologically transitive (\textit{i.e.}, indecomposability of the system)\\
(for any pair of open sets $U,V \subset \mathcal{X}$, $\exists k > 0 . 
f^k (U) \cap V \neq \emptyset$)
\\
\onslide<2>{\alert<2>{Addressed property: preimage resistance}}.
\item  $f$ is regular (\textit{i.e.}, fundamentally different points coexist)\\
(the set of periodic points is dense in $\mathcal{X}$).
\item  $f$: sensitive dependent on initial conditions (SDIC)\\
($
\exists \delta  > 0 .  \forall x \in  \mathcal{X}
\textrm{ and } 
\forall V \textrm{ neighborhood of $x$}.
\exists y \in V \textrm{ and } 
\exists n \ge 0 . 
d(f^n(x); f^n(y))> \delta
$)\\
\onslide<2>{\alert<2>{Addressed properties: avalanche effect}}.
\end{enumerate}
\end{block}
\footnote{\bibentry{devaney}}