Journal of Statistical Physics, Vol. W, Nos. 3/4, IWX A Short Existence Proof for Correlation Dimension Anthony Manning1 and Karoly Simon2 Received October 2K, 7997 The Grassberger Hentschel Procaccia correlation dimension has been put on a rigorous basis by Pesin and Tempelman. We simplify their proof that this dimension is given in terms of the measure of neighborhoods of the diagonal. KEY WORDS: Correlation dimension; ergodic measure; neighborhood of diagonal. Let (X, p) be a separable metric space. Suppose that n is an ergodic prob- ability measure for the continuous map/: X-> X. The r neighbourhood of the diagonal in XxX is denoted by Sr. That is Sr := {(.v, v)e Xx X: p(x,y)^r}. The function q>(r) = \'(Sr) is monotone increasing where v is the product measure /*x/;. For xeX and neN we let C(.v, n, r) denote l/n2${(i, j): (/''(x), fJ(x))eSr,0^i, j<n}, the proportion of pairs of points in part of the orbit that are closer than r. Roughly speaking, if, for /i almost every x, for large n and small r, we have C(,v, n, r) ~ ra then a is called the correlation dimension' 3' of the measure //. To give a precise definition of the correlation dimension it is fundamental to prove the following theorem, as was done for invertible / by Pesin' u and, in a more general context, by Pesin and Tempelman. <2> Theorem 1. There is a set YcX of full ^-measure such that for each xe Y provided (p is continuous at r. 1 Mathematics Institute, University of Warwick, Coventry, CV4 7AL, U.K. 2 Institute of Mathematics, University of Miskolc, Miskolc-Egyetemvaros H-3515, Hungary. 1047 0022-4715/98/0200-1047115.00/0 © 1998 Plenum Publishing Corporation