The strong open set condition in the random case

Abstract

Summary: To describe some fractal properties of a self-similar set ormeasure, such as the Hausdorff dimension and the multifractal spectrum, itis useful that it satisfy the strong open set condition, which means thereis an open set satisfying the open set condition and, additionally, a partof the self-similar set must meet the open set. It is known that in thenon-random case the strong open set condition and the open set condition areequivalent. This paper treats the random case. If the open set condition isassumed, we show that there is a random open set satisfying the strong openset condition. Further, we give an application to multifractal analysis ofthe random self-similar fractal.