Arnold cat map, Ulam method and time reversal

LEONARDO ERMANN; DIMA L. SHEPELYANSKY

PHYSICA D - NONLINEAR PHENOMENA

ELSEVIER SCIENCE BV

Lugar: Amsterdam; Año: 2012 vol. 241 p. 514 - 518

0167-2789

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