ON THE EXACT DIMENSION OF MANDELBROT MEASURE
Abstract: We develop, in the context of the boundary of a supercritical Galton–Watson tree, a
uniform version of the argument used by Kahane (1987) on homogeneous trees to estimate
almost surely and simultaneously the Hausdorff and packing dimensions of the Mandelbrot
measure over a suitable set . As an application, we compute, almost surely and
simultaneously, the Hausdorff and packing dimensions of the level sets of infinite
branches of the boundary of the tree along which the averages of the branching random walk
have a given limit point.
2000 AMS Mathematics Subject Classification: Primary: 11K55; Secondary:
Keywords and phrases: Mandelbrot measure, Hausdorff dimension.