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+\documentclass[cvs,envcountsect]{svjour}
+\usepackage{amsfonts}
+\usepackage{psfig}
+
+\begin{document}
+
+
+\newcommand{\Ref}[1]{(\ref{#1})}
+\newcommand{\R}{{\mathbb R}}
+\newcommand{\N}{{\mathbb N}}
+\newcommand{\cB}{{\cal B}}
+\newcommand{\cM}{{\cal M}}
+\newcommand{\diam}{\mathop{\rm diam}\nolimits}
+
+\renewcommand{\labelenumi}{(\roman{enumi})}
+
+
+\title{An adaptive subdivision technique
+ for the approximation\\ of
+ attractors and invariant measures}
+
+\author{Michael Dellnitz\thanks{Research of the authors is partly supported
+ by the Deutsche Forschungsgemeinschaft under Grant De 448/5-2
+ and by the Konrad-Zuse-Zentrum f\"ur Informationstechnik Berlin.},
+ Oliver Junge}
+
+\institute{Mathematisches Institut, Universit\"at Bayreuth,
+ D-95440 Bayreuth, Germany\\
+ {\scriptsize\tt
+ (http://www.uni-bayreuth.de/departments/math/\~{}mdellnitz;
+ http://www.uni-bayreuth.de/departments/math/\~{}ojunge)}}
+
+\date{Received: 13 January 1997 / Accepted: 29 September 1997 \\[1em]
+ Communicated by: P. Deuflhard}
+
+\maketitle
+
+\begin{abstract}
+Recently subdivision techniques have been introduced in the
+numerical investigation of the temporal behavior of
+dynamical systems. In this article we intertwine the subdivision
+process with the computation of invariant measures and propose an
+adaptive scheme for the box refinement which is based on the
+combination of these methods. Using this new algorithm the numerical
+effort for the computation of box coverings is in general significantly
+reduced, and we illustrate this fact by several numerical examples.
+\end{abstract}
+
+\section{Introduction}
+%
+Recently subdivision methods have been successfully applied to the
+numerical analysis of complex dynamical behavior
+(e.g.\ \cite{Eiden,DH1,DH2,DJ:96,DJchua:97}). These methods can be used for two
+essentially different purposes: the first is to
+understand the geometric structure of an underlying attractor.
+Secondly the goal may be to approximate the observable dynamical behavior
+of the underlying system in a specific region of state space by
+the computation of invariant measures. This paper
+concerns the second possibility, and we propose an adaptive scheme
+incorporated into the subdivision technique which allows to
+reduce the numerical effort significantly.
+
+Obviously the dynamical behavior
+just needs to be approximated on the support of a certain invariant measure.
+Indeed, the idea for the adaptive principle stated here
+is to intertwine the subdivision techniques with the computation of a
+natural invariant measure, an {\em SBR-measure}, say. Roughly speaking,
+the size of the covering boxes is reduced in those parts of
+state space where the natural invariant measure $\mu$ is concentrated, and,
+on the other hand, boxes are not subdivided in areas which have
+$\mu$-measure zero.
+
+The main goal of this article is to illustrate the efficiency of
+the new method by numerical examples. For that purpose we
+consider several dynamical systems
+for which the SBR-measure is known analytically since this allows us to
+compare the numerical results obtained by the {\em adaptive\/}
+subdivision algorithm to those obtained by the {\em standard\/}
+subdivision procedure. The adaptive algorithm is essentially based
+on the combination of two existing methods for which
+convergence results are known. However, this fact does not
+immediately imply convergence of the adaptive method as well.
+Rather this theoretical but relevant problem is
+currently under investigation, and the results will be published elsewhere.
+
+An outline of the paper is as follows: in Sect.\,\ref{sec:CSA} we
+recall the standard subdivision technique from \cite{DH1}. The numerical method
+for the approximation of SBR-measures is described in Sect.\,\ref{sec:fp}.
+Then, in Sect.\,\ref{sec:ASA}, we present our adaptive subdivision technique, and
+the efficiency of this method is illustrated by several examples in
+Sect.\,\ref{sec:num}.
+
+
+\section{The standard subdivision algorithm}
+\label{sec:CSA}
+%
+The purpose is to approximate invariant sets of discrete
+dynamical systems of the form
+\[
+x_{j+1} = f(x_j),\quad j=0,1,\ldots,
+\]
+where $f$ is a continuous mapping on $\R^n$.
+The central object which is approximated by the subdivision algorithm
+developed in \cite{DH1} is the so-called {\em relative global attractor},
+\begin{equation}
+\label{eq:relativAttractor2}
+A_Q=\bigcap_{j\geq0}f^j(Q),
+\end{equation}
+where $Q\subset\R^n$ is a compact subset. Roughly speaking,
+the set $A_Q$ should be viewed as the {\em union of unstable
+manifolds of invariant objects inside $Q$}. In particular,
+$A_Q$ may contain subsets of $Q$ which cannot be approximated
+by direct simulation.
+
+The subdivision algorithm for the approximation of $A_Q$
+generates a sequence $\cB_0,\cB_1,\cB_2,\ldots$ of finite
+collections of boxes with the property that for all
+integers $k$ the set $Q_k=\bigcup_{B\in\cB_k}B$
+is a covering of the relative global attractor under consideration.
+Moreover the sequence of coverings
+is constructed in such a way that the diameter of the boxes,
+\[
+\diam(\cB_k) = \max_{B\in\cB_k}\diam(B)
+\]
+converges to zero for $k\rightarrow\infty$.
+
+Given an initial collection $\cB_0$, one inductively obtains $\cB_k$ from
+$\cB_{k-1}$ for $k=1,2,\ldots$ in two steps.
+\begin{enumerate}
+\item {\em Subdivision:} Construct a new collection $\hat\cB_k$ such that
+\begin{eqnarray*}
+\bigcup_{B\in\hat\cB_k}B &=& \bigcup_{B\in\cB_{k-1}}B \\
+\mbox{and}\quad
+\diam(\hat\cB_k) &\leq& \theta\diam(\cB_{k-1})
+\end{eqnarray*}
+for some $0<\theta<1$.
+\item {\em Selection:} Define the new collection $\cB_k$ by
+\[
+\cB_k=\left\{B\in\hat\cB_k \!:\! f^{-1}(B)\!\cap\!\hat B\!\ne\!\emptyset
+\!\!\quad\mbox{for some}\!\!\quad\hat B\!\in\!\hat\cB_k\right\}.
+\]
+\end{enumerate}
+
+The following proposition establishes a general convergence property of
+this algorithm.
+
+\begin{proposition}[\cite{DH1}]
+\label{prop:sdconv}
+Let $A_Q$ be the global attractor relative to the compact set $Q$,
+and let $\cB_0$ be a finite collection of closed subsets with
+$Q_0=Q$. Then
+\[
+\lim\limits_{k\rightarrow\infty}
+h\left( A_Q, Q_k \right)=0,
+\]
+where we denote by $h(B,C)$ the usual Hausdorff distance between two
+compact subsets $B,C\subset\R^n$.
+\end{proposition}
+
+\setcounter{equation}{0}
+\section{Approximation of SBR-measures}
+\label{sec:fp}
+%
+Recently it has been shown in \cite{DJ:96} how to compute numerically
+approximations of an {\em SBR-measure\/} supported on a hyperbolic
+invariant set. Since we want to use this method in our adaptive scheme
+we now sketch the main ingredients of this algorithm.
+To make the ideas more transparent we simplify the description
+drastically by avoiding all technical details
+concerning the underlying mathematical foundation in Ergodic Theory.
+
+The crucial observation is that the calculation of
+invariant measures can be viewed as a fixed point problem.
+Let ${\cal M}$ be the set of probability measures on $\R^n$.
+Then $\mu\in{\cal M}$ is invariant if and
+only if it is a fixed point of the {\em Frobenius-Perron operator\/}
+$P:{\cal M}\to {\cal M}$,
+\begin{equation}
+\label{eq:fp-fixed}
+(P\mu)(B)=\mu(f^{-1}(B))\quad\mbox{for all measurable $B\subset\R^n$}.
+\end{equation}
+
+For a discretization of the operator $P:\cM\rightarrow\cM$ we
+replace $\cM$ by a finite dimensional
+set $\cM_k$: let $B_i\in\cB_k$, $i=1,\ldots,N$,
+denote the boxes in the covering obtained after $k$ steps in the subdivision
+algorithm and set as before $Q_k=\bigcup_{B\in\cB_k}B$.
+We choose $\cM_k$ to be the set of ``discrete probability
+measures'' on $\cB_k$, that is,
+\[
+\cM_k = \left\{ u:\cB_k\to [0,1] \quad\Bigg|
+\quad \sum_{i=1}^N u(B_i)=1\right\}.
+\]
+Assuming that $f(Q_k)\subset Q_k$ the discretized
+Frobenius-Perron operator $P_k:\cM_k\rightarrow\cM_k$ is given by
+\begin{eqnarray}\label{eq:FP}
+v= P_k u,\quad v(B_i)=\sum_{j=1}^N
+\frac{m(f^{-1}(B_i)\cap B_j)}{m(B_j)}u(B_j),\\
+\hfill i=1,\ldots,N,\nonumber
+\end{eqnarray}
+where $m$ denotes Lebesgue measure. Now a fixed point $u=P_k u$ of
+$P_k$ provides an approximation to an invariant measure of $f$.
+\begin{remark}
+For the mathematically precise statement on the convergence of this
+method one would have to introduce the concept of {\em small random
+perturbations}. The reason is that this allows one to use a result of
+Yu.\ Kifer on the convergence of invariant measures in the perturbed
+systems to the SBR-measure (\cite{Kifer:86}).
+However, the purpose of this article is to develop and to test an adaptive
+scheme for the box refinement in the subdivision algorithm rather than to
+explain the theoretical background concerning the computation of
+SBR-measures. Therefore the
+reader is referred to \cite{DJ:96} for the rigorous mathematical
+treatment.
+\end{remark}
+
+
+\section{The adaptive subdivision algorithm}
+\label{sec:ASA}
+%
+As mentioned above, the standard subdivision algorithm may approximate
+a part of the global attractor which is dynamically irrelevant in the
+sense that no invariant measure has support on this subset.
+The reason is that {\em each\/} box is subdivided in a step of the
+subdivision algorithm regardless of any information on the dynamical
+behavior. In particular, also those subsets of the relative global
+attractor corresponding to unstable or transient dynamical behavior
+are approximated by the standard procedure.
+
+On the other hand, if one is mainly interested in the approximation of
+the support of the (natural) invariant measure rather than in the precise
+geometric structure of the global attractor then this strategy may lead to
+unnecessary high storage and computation requirements. In the following
+we present a modified subdivision strategy which avoids this drawback:
+roughly speaking,
+\begin{itemize}
+\item[--] in the subdivision step we use
+the information on the actual approximation of the
+invariant measure to decide whether or not a box should be subdivided;
+\item[--] in the selection step we keep only those boxes
+which have a nonempty intersection with the support of the invariant measure.
+\end{itemize}
+
+To be more precise, let $\{\delta_k\}$ be a sequence of positive
+real numbers such that $\delta_k\to 0$ for $k\to\infty$. The algorithm
+generates a sequence of pairs
+\[
+(\cB_0, u_0),(\cB_1, u_1), (\cB_2, u_2), \ldots
+\]
+where the $\cB_k$'s are finite collections of compact subsets of
+$\R^n$ and the discrete measures $u_k:\cB_k\to[0,1]$ can be interpreted
+as approximations to the SBR-measure $\mu_{SBR}$:
+\[
+u_k(B) \approx \mu_{SBR}(B)\quad\mbox{for all $B\in\cB_k$}.
+\]
+
+Given an initial pair $(\cB_0, u_0)$, one inductively
+obtains $(\cB_k, u_k)$ from $(\cB_{k-1}, u_{k-1})$ for
+$k=1,2,\ldots$ in three steps:
+\begin{enumerate}
+\item {\em Subdivision:} Define
+\begin{eqnarray*}
+\cB_{k-1}^- &=&\{B\in\cB_{k-1}: u_{k-1}(B) < \delta_{k-1}\} \\
+\mbox{and}\qquad && \\ \cB_{k-1}^+ &=& \cB_{k-1}\backslash\cB_{k-1}^-.
+\end{eqnarray*}
+Construct a new (sub-)collection $\hat\cB_k^+$ such that
+\[
+\bigcup_{B\in \hat\cB_k^+} B = \bigcup_{B\in \cB_{k-1}^+} B
+\]
+where
+\[
+\diam(\hat\cB_k^+) \leq \theta\diam(\cB_{k-1}^+)
+\]
+for some $0 < \theta < 1$.
+\item {\em Calculation of the invariant measure:}
+Set
+\[
+\hat \cB_k = \cB_{k-1}^-\cup\hat\cB_k^+.
+\]
+For the collection $\hat \cB_k$ calculate the approximating
+invariant measure as the fixed point $\hat u_k$ of the discretized
+Frobenius-Perron operator defined by \Ref{eq:FP}.
+\item {\em Selection:} Set
+\[
+\cB_k = \{ B\in \hat \cB_k: \hat u_k(B) > 0\}
+\]
+and
+\[
+u_k = \hat u_k|_{\cB_k}.
+\]
+\end{enumerate}
+
+\begin{remark}\label{rmk:real}
+\begin{itemize}
+\item[(a)] In the realization of the algorithm we typically subdivide the
+boxes in the collection $\cB_k^+$ by bisection. This guarantees that
+the number of boxes is not growing too fast. For the details concerning the
+implementation the reader is again referred to \cite{DH1,DJ:96}.
+\item[(b)] In principle there is some freedom in choosing the sequence
+$\{\delta_k\}$ of positive numbers used in the subdivision step. Note however
+that this sequence determines the number of boxes which will be subdivided
+and hence it has a significant influence on the storage requirement.
+In the computations it turned out to be quite efficient to choose the
+average
+\[
+\delta_k = {1\over N_k}\sum_{B\in\cB_k} u_k(B) = {1\over N_k},
+\]
+where $N_k$ is the number of boxes in $\cB_k$.
+\item[(c)] In the numerical realization of the selection step (iii)
+we check whether $\hat u_k(B) > \epsilon$ where $\epsilon>0$ is chosen
+sufficiently small with respect to the machine precision.
+\end{itemize}
+\end{remark}
+
+
+\section{Numerical examples}
+\label{sec:num}
+%
+In this section we illustrate the efficiency of the adaptive scheme by
+several numerical examples. First we consider three one-dimensional
+mappings for which the SBR-mea\-sures are known analytically. For these
+cases we
+will see that, as expected, the new technique is particularly useful
+if the underlying invariant density has singularities. Additionally we
+consider the H\'{e}non map as a two-dimensional
+example and show the box refinement produced
+by the adaptive subdivision algorithm at a certain step.
+
+Before proceeding let us indicate some details concerning the implementation
+of the adaptive subdivision algorithm:
+\begin{itemize}
+\item[(a)] The subdivision is always done by bisection and the boxes
+are stored in a binary tree. This way we keep the storage requirement
+at a low level.
+\item[(b)] For the computation of the transition probabilities
+\linebreak $m(f^{-1}(B_i)\cap B_j)$ in \Ref{eq:FP}
+we use an exhaustion technique as described in \cite{GDK}. This method
+is particularly useful when -- as in our examples -- local Lipschitz constants
+are available for the underlying dynamical system.
+\item[(c)] The computation of the discrete measures is done by an inverse
+power method. In the solution of the corresponding linear systems
+the fact is taken into account that the discretized
+Frobenius-Perron operator is extremely sparse.
+\end{itemize}
+The adaptive algorithm is integrated into the C++ code {\sf GAIO}
+({\bf G}lobal {\bf A}nalysis of {\bf I}nvariant {\bf O}bjects).
+A link to a detailed description of {\sf GAIO} can be found on
+the homepages of the authors.
+
+
+\subsection*{Three one-dimensional examples}
+\label{subsec:1D}
+%
+Motivated by the numerical investigations in \cite{DingDuLi:93}
+we apply the adaptive subdivision algorithm to three different
+one-dimensional dynamical systems on the interval $[0,1]$. In each
+case we have chosen the initial collection $\cB_0=\{[0,1]\}$.
+
+\begin{itemize}
+\item[{\it 1.}]
+As a first example we consider the Logistic Map $f_1:[0,1]\to [0,1]$,
+\[
+f_1(x)=\lambda x(1-x)
+\]
+for $\lambda=4$. The unique absolutely continuous invariant measure $\mu$
+of $f_1$ has the density
+\[
+h_1(x)={1\over \pi \sqrt{x(1-x)}}
+\]
+(see e.g.\ \cite{Lasota:94}). Using the standard and the adaptive
+algorithm we have approximated this density on several levels
+and the results are shown in Table 1. We remark that even
+the computation for $\ell=20$ only takes about 50 sec
+on an MIPS R4400 cpu.
+
+\begin{table} \label{tab:log}
+\caption{Comparison between the standard and the adaptive subdivision
+algorithm for the Logistic Map. The minimal box volume in each row
+is $2^{-\ell}$}
+\begin{tabular}{ l l l l l }
+\hline
+$\ell$ &
+\multicolumn{2}{c}{number of boxes} & \multicolumn{2}{c}{$L^1$-error} \\
+ & standard & adaptive & standard & adaptive \\ \hline
+6 & 64 & 17 & 0.1390 & 0.1431 \\
+8 & 256 & 34 & 0.0670 & 0.0765 \\
+12 & 4096 & 151 & 0.0210 & 0.0258 \\
+16 & 65536 & 679 & 0.0064 & 0.0073 \\
+20 & $2^{20}$ & 2831 & - & 0.0021 \\
+\hline
+\end{tabular}
+\end{table}
+
+In Fig.~\ref{fig:log} we illustrate the fact that the size of the boxes
+is much smaller for those which are close to zero or one. Indeed, this is
+what we expect since the density has singularities in these points.
+
+\begin{figure*}
+\vbox{\hbox to\textwidth{\psfig{figure=cvs21.f1a,width=6cm}\hspace{5mm}
+\psfig{figure=cvs21.f1b,width=6cm}\hfill \parbox[b]{50mm}{%
+\caption[]{Illustration of the relation between the density and the
+ actual box refinement produced by the adaptive subdivision
+ algorithm for $\ell =10$: {\bf a} the density $h_1$;
+ {\bf b} the radii versus the midpoints of
+boxes}\vspace{2mm}}}}
+\label{fig:log}
+\end{figure*}
+
+\item[{\it 2.}] We consider the map $f_2:[0,1]\to [0,1]$,
+\renewcommand{\arraystretch}{2.1}
+\[
+f_2(x)=\left\{
+\begin{array}{ll}
+\displaystyle{2x\over 1-x^2} & \mbox{for}\quad 0\le x< \sqrt{2}-1, \\
+\displaystyle{1-x^2\over 2x} & \mbox{for}\quad \sqrt{2}-1 \le x \le 1.
+\end{array}
+\right.
+\]
+\renewcommand{\arraystretch}{1.0}
+Its invariant density is
+\[
+h_2(x)= {4\over \pi (1+x^2)}\; ,
+\]
+see Fig.\,\ref{fig:tent}.
+
+\begin{figure*}
+\vbox{\hbox to\textwidth{\psfig{figure=cvs21.f2a,width=6cm}\hspace{5mm}%
+\psfig{figure=cvs21.f2b,width=6cm}\hfill
+\parbox[b]{50mm}{%
+\caption[]{{\bf a} The map $f_2$; and {\bf b} its invariant density
+ $h_2$}\vspace{1mm}}}}
+\label{fig:tent}
+\end{figure*}
+
+In Table \ref{tab:tent} we present the numerical results
+for this case. As expected the application of the adaptive
+subdivision algorithm is not more efficient than the
+standard one since the invariant measure is quite close to Lebesgue
+measure.
+
+\begin{table}
+\caption[]{Comparison of the numerical results for $f_2$
+ ($\ell$ as in Table~1)}
+\label{tab:tent}
+\begin{tabular}{ l l l l l }
+\hline
+$\ell$ &
+\multicolumn{2}{c}{number of boxes} & \multicolumn{2}{c}{$L^1$-error} \\
+ & standard & adaptive & standard & adaptive \\ \hline
+6 & 64 & 45 & 0.0027 & 0.0053 \\
+8 & 256 & 187 & $6.5\cdot 10^{-4}$ & 0.0013 \\
+10 & 1024 & 759 & $1.7\cdot 10^{-4}$ & $3.2\cdot 10^{-4}$ \\
+12 & 4096 & 3047 & $4.6\cdot 10^{-5}$ & $7.8\cdot 10^{-5}$ \\
+\hline
+\end{tabular}
+\end{table}
+
+\item[{\it 3.}] Finally we consider the map $f_3:[0,1]\to [0,1]$,
+\[
+f_3(x) = \left({1\over 8} -2 \left|x-{1\over 2}\right|^3\right)^{1\over 3}
++ {1\over 2},
+\]
+with the invariant density
+\[
+h_3(x) = 12\left(x-{1\over 2}\right)^2.
+\]
+The graphs of $f_3$ and $h_3$ are shown in Fig.\,\ref{fig:stahl}.
+
+\begin{figure*}
+\vbox{\hbox to\textwidth{\psfig{figure=cvs21.f3a,width=6cm}\hspace{5mm}%
+\psfig{figure=cvs21.f3b,width=6cm}\hfill
+\parbox[b]{50mm}{%
+\caption[]{{\bf a} The map $f_3$; and {\bf b} its invariant density
+ $h_3$}\vspace{2mm}}}}
+\label{fig:stahl}
+\end{figure*}
+
+\begin{figure*}
+\vbox{\hbox to\textwidth{\psfig{figure=cvs21.f4a,height=5.3cm}\hspace{5mm}%
+\psfig{figure=cvs21.f4b,height=5.3cm}\hfill}}
+\caption[]{{\bf a} A tiling of the square $[-2,2]^2$ obtained by the
+ adaptive subdivision algorithm; and {\bf b} the subcollection
+ $\tilde{\cB}$ of boxes with discrete density bigger than
+ $0.35$ (see also \protect\Ref{eq:dm})}
+\label{fig:henon}
+\end{figure*}
+
+We now discuss the numerical results presented in Table~\ref{tab:stahl}.
+Note that the derivative of $f_3$ has singularities at two points
+inside $[0,1]$. This is the reason why several boxes get lost in
+the realization of the selection step in the standard subdivision
+algorithm. Consequently the computation of the invariant measure
+does not lead to satisfying results. In contrast to this no boxes are lost
+in the application of the adaptive subdivision technique, and accordingly
+the $L^1$-error is decreasing with an increasing number of subdivision steps.
+
+\begin{table}
+\caption[]{Comparison of the numerical results for $f_3$
+ ($\ell$ as in Table~1)}
+\label{tab:stahl}
+\begin{tabular}{ l l l l l }
+\hline
+$\ell$ &
+\multicolumn{2}{c}{number of boxes} & \multicolumn{2}{c}{$L^1$-error} \\
+ & standard & adaptive & standard & adaptive \\ \hline
+6 & 63 & 11 & 0.0260 & 0.2931 \\
+8 & 249 & 30 & 0.0105 & 0.2583 \\
+10 & 993 & 189 & 0.0110 & 0.0435 \\
+12 & 3967 & 816 & 0.0133 & 0.0065 \\ \hline
+\end{tabular}
+\end{table}
+\end{itemize}
+
+
+\subsection*{The H\'enon map}
+\label{subsec:henon}
+%
+We apply the adaptive subdivision algorithm to a two-di\-men\-sio\-nal
+example, namely a scaled version of the well known {\em H\'enon map}
+\[
+f:\R^2\to \R^2,\quad
+f(x,y) = (1-ax^2+y/5,\; 5bx).
+\]
+In the computations we have fixed the parameters by $a=1.2$, $b=0.2$,
+and we have chosen $\cB_0=\{[-2,2]^2\}$.
+
+In Fig.\,\ref{fig:henon} we present a tiling of the square $[-2,2]^2$
+obtained by the adaptive subdivision algorithm after several subdivision
+steps. The resulting box-collection $\cB$ consists of the
+grey boxes shown in part (a) of this figure.
+We expect that due to the numerical approximation some boxes have
+positive {\em discrete\/} measure although they do not intersect the
+support of the {\em real\/} natural invariant measure.
+Having this in mind we neglect those boxes with very small discrete
+measure and show in Fig.\,\ref{fig:henon}b a subcollection
+$\tilde\cB\subset\cB$ with the property that
+\begin{equation}
+\label{eq:dm}
+\sum_{B\in\tilde\cB} u(B) \approx 0.99
+\end{equation}
+(see also Remark~\ref{rmk:real}(c)).
+
+\begin{figure}
+\vbox{\hbox to\textwidth{\psfig{figure=cvs21.f5,width=8.6cm}\hfill}}
+\caption[]{Illustration of the (natural) invariant measure for the
+ H\'enon map. The picture shows the density of the discrete
+ measure on $\tilde\cB$, see \protect\Ref{eq:dm}}
+\label{fig:hen2}
+\end{figure}
+
+\begin{remark}
+For our choice of the parameter values we cannot explicitly
+write down a natural invariant measure. Hence it is impossible
+to compare our numerical results using analytical ones.
+Moreover, it is not even known for an arbitrary choice of the parameter
+values whether or not the H\'enon map possesses an SBR-measure.
+However, recently it was proved by M.\ Benedicks and L.-S.\ Young that
+the H\'enon map indeed has an SBR-measure for a
+``large'' set of parameter values, see \cite{BenYoung}.
+\end{remark}
+
+
+Finally we apply the numerical techniques described in
+\cite{DJ:96} to determine the essential dynamical behavior of the
+H\'enon map for this choice of parameter values. An approximation
+of the (natural) invariant measure obtained by the adaptive
+subdivision algorithm is shown in Fig.\,\ref{fig:hen2}. This
+computation is based on the total number of 1514 boxes inside the
+square $[-2,2]^2$, whereas the support of the
+invariant measure is covered by 1442 boxes.
+The results indicate that the H\'enon map exhibits complicated dynamical
+behavior.
+
+Moreover it can be shown that the areas
+where the density is colored red resp.\ blue
+are permuted cyclically by the mapping. Hence altogether
+we may conclude that for these parameter values the H\'enon
+map exhibits a two-cycle (the ``macro-dynamics'') in addition to
+unpredictable (chaotic) behavior. This fact is also demonstrated
+by a {\sf Java}-animation for which a link can be found on the
+homepages of the authors. \thisbottomragged
+
+
+
+\begin{thebibliography}{10.}
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+\newpage
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+\end{thebibliography}
+
+\end{document}