Anosov Properties of a Symplectic Map with Time-Reversal Symmetry

TL;DR

This paper analyzes a symplectic map with time-reversal symmetry, demonstrating its mixing properties, uniqueness of the invariant measure, and positive entropy, highlighting its chaotic behavior.

Contribution

It introduces a specific symplectic map derived from a Hamiltonian that exhibits time-reversal symmetry and proves its mixing and chaotic properties analytically.

Findings

Initial density converges to a uniform distribution

Unique SRB and physical measure established

Kolmogorov-Sinai entropy is positive

Abstract

This study presents a specific symplectic map, derived from a Hamiltonian, as a model that exhibits time-reversal symmetry on a microscopic scale. Based on the analysis, any initial density function, defined almost everywhere, converges to a uniform distribution in terms of mixing (irreversible behavior) on a macroscopic level. Furthermore, we established that this mixing invariant measure is a unique equilibrium state, unique SRB measure, and physical measure. Additionally, through analytical proof, we have shown that the Kolmogorov-Sinai entropy, representing the average information gain per unit time is positive. This was achieved by validating the Pesin's formula and demonstrating that the critical exponent of the Lyapunov exponent is $1/2$.