  We study numerically multifractal properties of two models of one-dimensional
quantum maps, a map with pseudointegrable dynamics and intermediate spectral
statistics, and a map with an Anderson-like transition recently implemented
with cold atoms. Using extensive numerical simulations, we compute the
multifractal exponents of quantum wave functions and study their properties,
with the help of two different numerical methods used for classical
multifractal systems (box-counting method and wavelet method). We compare the
results of the two methods over a wide range of values. We show that the wave
functions of the Anderson map display a multifractal behavior similar to
eigenfunctions of the three-dimensional Anderson transition but of a weaker
type. Wave functions of the intermediate map share some common properties with
eigenfunctions at the Anderson transition (two sets of multifractal exponents,
with similar asymptotic behavior), but other properties are markedly different
(large linear regime for multifractal exponents even for strong
multifractality, different distributions of moments of wave functions, absence
of symmetry of the exponents). Our results thus indicate that the intermediate
map presents original properties, different from certain characteristics of the
Anderson transition derived from the nonlinear sigma model. We also discuss the
importance of finite-size effects.
