Bounds for the Lq-spectra of a self-similar multifractal not satisfying the Open Set Condition

By now the Lq-spectra of self-similar measures satisfying the so-called Open Set Condition is well understood. However, if the Open Set Condition is not satisfied, then almost nothing is known. In this paper we provide non-trivial bounds for the Lq-spectra of an arbitrary self-similar measure. We emphasize that we are considering arbitrary self-similar measures (and sets) which are not assumed to satisfy the Open Set Condition or similar separation conditions.