  We consider the cumulant generating function of the logarithm of the distance
between two infinitesimally close trajectories of a chaotic system. Its
long-time behavior is given by the generalized Lyapunov exponent $\gamma(k)$
providing the logarithmic growth rate of the $k-$th moment of the distance. The
Legendre transform of $\gamma(k)$ is a large deviations function that gives the
probability of rare fluctuations where the logarithmic rate of change of the
distance is much larger or much smaller than the mean rate defining the first
Lyapunov exponent. The only non-trivial zero of $\gamma(k)$ is at minus the
correlation dimension of the attractor which for incompressible flows reduces
to the space dimension. We describe here general properties constraining the
form of $\gamma(k)$ and the Gallavotti-Cohen type relations that hold when
there is symmetry under time-reversal. This demands studying joint growth rates
of infinitesimal distances and volumes. We demonstrate that quartic polynomial
approximation for $\gamma(k)$ does not violate the Marcinkiewicz theorem on
invalidity of polynomial form for the generating function. We propose that this
quartic approximation will fit many experimental situations, not having the
effective time-reversibility and the short correlation time properties of the
quadratic Grassberger-Procaccia estimates. We take the existing $\gamma(k)$ for
turbulent channel flow and demonstrate that the quartic fit is nearly perfect.
The violation of time-reversibility for the Lagrangian trajectories of the
incompressible Navier-Stokes turbulence below the viscous scale is considered.
We demonstrate how the fit can be used for finding the correlation dimensions
of strange attractors via easily measurable quantities. We provide a simple
formula via the Lyapunov exponents, holding in quadratic approximation, and
describe the construction of the quartic approximation.
