Spectra of stretching numbers and helicity angles in dynamical systems

We define a ''stretching number'' (or ''Lyapunov characteristic number for one period'') (or ''stretching number'')) a = ln \xi(t + 1)/xi(t)\ as the logarithm of the ratio of deviations from a given orbit at times t and t + 1. Similarly we define a ''helicity angle'' as the angle between the deviation xi(t) and a fixed direction. The distributions of the stretching numbers and helicity angles (spectral are invariant with respect to initial conditions in a connected chaotic domain. We study such spectra in conservative and dissipative mappings of 2 degrees of freedom and in conservative mappings of 3-degrees of freedom. In 2-D conservative systems we found that the lines of constant stretching number have a fractal form.