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%Grassberger and Procaccia$B$NO@J8(B
%Physical Review Letters 50(1983), 346-349.
%
\begin{flushright}
$BC+@n@6N4(B $BLu(B
\end{flushright}

\noindent
{\it Physical Review Letters} {\bf 50} (1983), 346-349

\vspace{0.5cm}
\begin{center}
{\Large
{\bf $B%9%H%l%s%8%"%H%i%/%?!<$NFCD'$E$1(B \\
Characterization of Strange Attractors} 
}

\vspace{0.3cm}
Peter Grassberger and Itamar Procaccia

\vspace{0.3cm}
Chemical Physica Department, Weizmann Institute of Science, \\ 
Rehovot, 76100, Israel
\end{center}

\vspace{0.5cm}
\begin{center}
{\bf $BMWLs(B} 
\end{center}

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\vspace{0.5cm}
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$B$3$N$h$&$JD9;~7ONs$N(B$n$$B8D$NE@$r(B
$\{ \overrightarrow{X}_i \}_{i=1}^N \equiv \{ \overrightarrow{X}(t + i \tau) \}_{i=1}^N$$B$GI=$o$9!%$3$3$G(B$\tau$$B$OG$0U$K8GDj$7$?;~4VA}J,$G$"$k!%Aj4X@QJ,$NDj5A$O<!$N$H$*$j$G$"$k!%(B
$$ \begin{array}{ll}

  C(r) \equiv & \displaystyle 
\lim_{N \rightarrow \infty} \frac{1}{N^2} \sum_{i,j=1}^N 
\theta(r - |\overrightarrow{X}_i - \overrightarrow{X}_j|) \\
& \equiv \displaystyle \int_0^r d^d r' c(\overrightarrow{r}'),
   \end{array}   \eqno(1)
$$
$B$3$3$G(B$\theta(x)$$B$O%X%S%5%$%I4X?t$G$"$j!"(B$c(\overrightarrow{r})$$B$OI8=`E*Aj4X4X?t$G$"$k!%$3$NO@J8$N<gMWE@$O(B$r$$B$,>.$5$$$H$-(B$C(r)$$B$,(B$r$$B$N$Y$-$N$h$&$K$U$k$^$&$3$H$G$"$k!%$9$J$o$A!"(B
$$
  C(r) \propto r^{\nu} .              \eqno(2)
$$
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$$
  M(l) \sim l^{-D} .   \eqno(3)
$$
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$$
  C(l) \sim \displaystyle 
\frac{1}{N^2} \sum_{i=1}^{M(l)} \mu_i^2 = \displaystyle 
\frac{M(l)}{N^2} \langle \mu^2 \rangle .   \eqno(4)
$$
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$$ \begin{array}{ll}
  C(l) \geq \displaystyle 
\frac{M(l)}{N^2} \langle \mu^2 \rangle =  & \displaystyle 
\frac{1}{N^2 M(l)} \left[ \sum_{i=1}^{M(l)} \mu_i \right]^2 \\
 & = \displaystyle \frac{1}{M(l)} \\
 & \sim l^D .
   \end{array}    \eqno(5)
$$
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$$
  \nu \leq D .    \eqno(6)
$$

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$$
  S(l) = - \sum_{i-1}^{M(l)} p_i \ln p_i ,   \eqno(7)
$$
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$$
  S(l) = S_0 - \sigma \ln l ,            \eqno(8)
$$
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$$
  \sigma \leq D .   \eqno(9)
$$

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$$
  C(l) \sim \sum_{i=1}^{M(l)} p_i^2 = 
\sum_{j=1}^{M(2l)} P_j^2 \sum_{i \in j} \omega_i^2 . 
$$
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\langle \omega \ln \omega \rangle / \langle \omega \rangle . 
$$
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$$
  \nu \leq \sigma \leq D .              \eqno(10)
$$

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$$
  \overrightarrow{\xi}_i = (X_i, X_{i+\tau}, X_{i+2 \tau}, \ldots, X_{i+ j \tau}) , 
$$
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\vspace{0.5cm}
{\bf $B;29MJ88%(B}

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$^{22}$$B$3$NO@J8$rEj9F$7$F$+$i(BL.S. Young$B$NJs9p!"(BJ.D. Farmer, E. Ott, and J.A. Yorke$B$NJs9p$rCN$C$?!%$+$l$i$O>.$5$$J}$N<\EY$K$D$$$F5DO@$7$?!%(B

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