Typical properties of correlation dimension.
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.
scientific article; zbMATH DE number 2084037