Arnold cat map, Ulam method and time reversal
arXiv:1107.0437 · doi:10.1016/j.physd.2011.11.012
Abstract
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.
7 pages, 8 figures. Research done at Quantware http://www.quantware.ups-tlse.fr/