  We study the properties of the Arnold cap map on a torus with a several
periodic sections using the Ulam method. This approach generates a Markov chain
with the Ulam matrix approximant. We study numerically the spectrum and
eigenstates of this matrix showing their relation with the Fokker-Plank
relaxation and the Kolmogorov-Sinai entropy. We show that, in the frame of the
Ulam method, the time reversal property of the map is preserved only on a short
Ulam time which grows only logarithmically with the matrix size. Parallels with
the evolution in a regime of quantum chaos are also discussed.
