Arnold cat map, Ulam method and time reversal

TL;DR

This paper investigates the Arnold cat map using the Ulam method, analyzing its spectral properties, time reversal behavior, and connections to quantum chaos, revealing that time reversal is only preserved briefly.

Contribution

It introduces a numerical analysis of the Arnold map with the Ulam method, highlighting the short Ulam time for time reversal and its relation to quantum chaos phenomena.

Findings

Time reversal is preserved only on a short Ulam time that grows logarithmically.

Spectral properties relate to Fokker-Planck relaxation and Kolmogorov-Sinai entropy.

Parallels are drawn between classical chaos and quantum chaos regimes.

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.