Linking Lyapunov exponents to low‑frequency spectral asymptotics

Derive rigorous mathematical relations connecting classical Lyapunov exponents to the low‑frequency asymptotics of spectral functions (and corresponding fidelity susceptibilities), thereby establishing how trajectory instability measures quantitatively determine long‑time observable fluctuations.

Background

The paper demonstrates numerically that, in mixed phase space, spectral functions averaged over chaotic trajectories exhibit non‑decaying low‑frequency tails (approximately 1/ω), while those averaged over regular trajectories decay as ω^2. This empirical link mirrors the separation seen via Lyapunov exponents.

However, an explicit mathematical derivation relating Lyapunov exponents to the low‑frequency spectral behavior and fidelity susceptibility is currently missing, leaving a gap between trajectory‑level instability measures and ensemble‑averaged dynamical response.