  We study the eigenstates of open maps whose classical dynamics is
pseudointegrable and for which the corresponding closed quantum system has
multifractal properties. Adapting the existing general framework developed for
open chaotic quantum maps, we specify the relationship between the eigenstates
and the classical structures, and quantify their multifractality at different
scales. Based on this study, we conjecture that quantum states in such systems
are distributed according to a hierarchy of classical structures, but are
multifractal instead of ergodic at each level of the hierarchy. This is visible
for sufficiently long-lived resonance states at scales smaller than the
classical structures. Our results can guide experimentalists in order to
observe multifractal behavior in open systems.
