The Arnold cat map, the Ulam method and time reversal

We study the properties of the Arnold cat map on a torus with 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–Planck 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.