Hard and soft multifractal processes

We show that multifractal notions encompass a wider variety of phenomena than often believed. Ranked by increasing highest order of singularities we have geometric, then microcanonical and finally canonical multifractals. They are respectively localized and “calm”, delocalized and “calm”, and delocalized and “wild”. Canonical multifractals may also involve rare violent (“hard”) singularities which cause high order statistical moments to diverge. We demonstrate this classification in the framework of universal multifractals characterized by three fundamental exponents.