We explore the exact packing dimension of certain random recursive constructions. In case of polynomial decay at 0 of the distribution function of random variable X, associated with the construction, we prove that it does not exist, and in case of exponential decay it is t α | log | log t|| β , where α is the fractal dimension of the limit set and 1/β is the rate of exponential decay.

We consider primitive aperiodic substitutions of constant length q and prove that, in order to have a Lebesgue component in the spectrum of the associated dynamical system, it is necessary that one of the eigenvalues of the substitution matrix equals √ q in absolute value. The proof is based on results of M. Queffélec, combined with estimates of the local dimension of the spectral measure at zero.

We discuss the definition and measurability questions of random fractals with infinite branching, and find, under certain conditions, a formula for the upper and lower Minkowski dimensions. For the case of a random self-similar set we obtain the packing dimension.Résumé. Nous discutons les questions de définition et de la mesurabilité des fractales aléatoires avec ramification infinie, et trouvons sous certaines conditions une formule pour les dimensions de Minkowski supérieure et inférieure. En cas d'ensemble aléatoire auto-similaire nous obtenons la dimension d'entassement.

We study porosities in the Mandelbrot percolation process. We show that, almost surely at almost all points with respect to the natural measure, the mean porosities of the set and the natural measure exist and are equal to each other for all parameter values outside of a countable exceptional set. As a corollary, we obtain that, almost surely at almost all points, the lower porosities of the set and the natural measure are equal to zero, whereas the upper porosities obtain their maximum values.