  In this Letter we show that the analysis of Lyapunov-exponents fluctuations
contributes to deepen our understanding of high-dimensional chaos. This is
achieved by introducing a Gaussian approximation for the large deviation
function that quantifies the fluctuation probability. More precisely, a
diffusion matrix $\bf D$ (a dynamical invariant itself) is measured and
analysed in terms of its principal components. The application of this method
to three (conservative, as well as dissipative) models, allows: (i) quantifying
the strength of the effective interactions among the different degrees of
freedom; (ii) unveiling microscopic constraints such as those associated to a
symplectic structure; (iii) checking the hyperbolicity of the dynamics.
