Dimensionality of stable and unstable directions in the gravitational N-body problem

The gravitational n-body problem is chaotic. Phase trajectories that start very near each other separate rapidly. The rate looks exponential over long times. At any instant, trajectories separated in certain directions move a part rapidly (unstable directions), while those separated in other directio ns stay about the same (stable directions). Unstable directions lie along e igenvectors that correspond to positive eigenvalues of the matrix of force gradients. The number of positive eigenvalues of that matrix gives the dime nsionality of stable regions. This number has been studied numerically in a series of 100-body integrations. It continues to change as long as the int egration continues because the matrix changes extremely rapidly. On average , there are about 1.2n unstable directions out of 3n. Issues of dimensional ity arise when the tools of ergodic studies are brought to bear on the prob lem of trajectory separation. A method of estimating the rate of trajectory separation based on matrix descriptions is presented in this note. Severe approximations are required.