Typical properties of correlation dimension. (English) Zbl 1048.37020

Myjak, Józef; Rudnicki, Ryszard

Real Anal. Exch. 28(2002-2003), No. 2, 269-278 (2003).

Consider a polish space \(X\), with metric \(\rho\). The family of all Borel probability measures on \(X\) is denoted by \(\mathcal M\). The authors consider two metrics on \(\mathcal M\): the supremum metric \(d_1\) and the Fortet-Mourier distance \(d_2\), defined as the supremum of all numbers \(| \int_X f\,d\mu_1-\int_Xf\,d\mu_2| \) where \(f\) ranges in the family of Lipschitz-continuous functions of uniform norm at most \(1\) and Lipschitz constant \(1\) (with respect to \(\rho\)). The upper and lower correlation dimensions of a measure \(\mu\) are the \(\lim\sup\) and \(\liminf\) of \({1\over\log r}\log\int_X\mu \bigl( B(x,r)\bigr)\,d\mu(x)\) as \(r\to0\), respectively.
The principal result of the paper states that 1) the upper correlation dimension is zero for all \(\mu\) in a residual in \((\mathcal M,d_1)\); 2) the lower correlation dimension is zero for all \(\mu\) in a residual in \((\mathcal M,d_2)\); and 3) the upper correlation dimension lies between two values dependent on the so-called entropy dimension of balls for all \(\mu\) in a residual in \((\mathcal M,d_2)\). Carefully chosen Cantor sets in the real line witness that the estimates obtained in 3) are sharp.

Reviewer: K. P. Hart (Delft)

MSC:

37C45	Dimension theory of smooth dynamical systems
54E52	Baire category, Baire spaces
28A33	Spaces of measures, convergence of measures

Keywords:

measure; correlation dimension; entropy dimension; residual set