  We study the properties of the Google matrix generated by a coarse-grained
Perron-Frobenius operator of the Chirikov typical map with dissipation. The
finite size matrix approximant of this operator is constructed by the Ulam
method. This method applied to the simple dynamical model creates the directed
Ulam networks with approximate scale-free scaling and characteristics being
rather similar to those of the World Wide Web. The simple dynamical attractors
play here the role of popular web sites with a strong concentration of
PageRank. A variation of the Google parameter $\alpha$ or other parameters of
the dynamical map can drive the PageRank of the Google matrix to a delocalized
phase with a strange attractor where the Google search becomes inefficient.
