Anosov properties of a symplectic map with time-reversal symmetry

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 Sinai-Ruelle-Bowen 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 Pesin's formula and demonstrating that the critical exponent of the Lyapunov exponent is 1/2.