## Typical properties of correlation dimension.(English)Zbl 1048.37020

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
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