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-rw-r--r--buch/papers/ifs/images/farn.eps2372
-rw-r--r--buch/papers/ifs/images/farncolor.eps2666
-rw-r--r--buch/papers/ifs/teil2.tex26
3 files changed, 5063 insertions, 1 deletions
diff --git a/buch/papers/ifs/images/farn.eps b/buch/papers/ifs/images/farn.eps
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index 0000000..597745b
--- /dev/null
+++ b/buch/papers/ifs/images/farn.eps
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+} stopped {handleerror} if
+ RawData flushfile
+} exec
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+} stopped {handleerror} if
+ RawData flushfile
+} exec
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diff --git a/buch/papers/ifs/teil2.tex b/buch/papers/ifs/teil2.tex
index a728340..8a7f76f 100644
--- a/buch/papers/ifs/teil2.tex
+++ b/buch/papers/ifs/teil2.tex
@@ -105,4 +105,28 @@ für jedes i mit einem $c_i < 1$. Dann existiert eine eindeutige kompakte Menge
\begin{equation}
F = \bigcup\limits_{i = 1}^{m} S_i(F)
\end{equation}
-TODO Text
+Weiter definieren wir die Transformation S auf kompakte Mengen ohne die leere Menge.
+\begin{equation}
+ S(E) = \bigcup\limits_{i = 1}^m S_i(E)
+\end{equation}
+Wird diese Transformation Iterativ ausgeführt, das heisst $S^0(E) = E, S^k(E) = S(S^{k-1}(E))$, und für jedes $i$ $S_i(E) \subset E$, gilt
+\begin{equation}
+ F = \bigcap\limits_{k = 1}^{\infty} S^k(E).
+\end{equation}
+In Worte gefasst bedeutet das, dass jede Gruppe von Kontraktionen iterativ ausgeführt, gegen eine eindeutige Menge konvergiert.
+Dies für jede Startmenge, solange diese ihre Transformierten wieder beinhaltet.
+Auf den Beweis wird verzichtet.
+\subsection{Beispiel: Barnsley-Farn}
+\begin{figure}
+ \label{ifs:farn}
+ \centering
+ \makebox[\textwidth][c]{
+ \includegraphics[width=1.4\textwidth]{papers/ifs/images/farn}}
+ \caption{Barnsley-Farn}
+\end{figure}
+\begin{figure}
+ \label{ifs:farncolor}
+ \centering
+ \includegraphics[width=0.7\textwidth]{papers/ifs/images/farncolor}
+ \caption{Vier Transformationen des Barnsley-Farn}
+\end{figure}