On Hausdorff dimension of random fractals (Q5943976)

Let \(\Delta\) denote the set of all finite sequences of natural numbers, and let \(M\) be a complete metric space. A random countable family of closed non-empty subsets of a \(M\) is a measurable map \({\mathbf I}\colon (\Omega,{\mathcal G},P)\to (2^M)^\Delta\) from a standard probability space to \(M\). Such a family is called a random construction if it is increasing (with regard to a natural ordering of finite sequences) and, almost surely in \(\omega\in\Omega\), the maximal diameter of all sets indexed with sequences of length \(n\) goes to \(0\) as \(n\to\infty\). Given a random construction \(I\), the corresponding random fractal is the random set \(K(\omega)\), the intersection taken over \(n=1,2,\dots\) of unions of images of all sequences of length \(n\). To every random construction the authors associate a family of linear operators in a finite dimensional space, depending on a real parameter and called the transition operator. The main result of the article under review asserts that the Hausdorff dimension of a random fractal almost surely equals the value of the parameter for which the spectral radius of the transition operator equals one. As an intermediate step in the proof, it is shown that the dimension in question equals the Hausdorff dimension of the Baire space equipped with a suitable random metric.