• confidence region
• hyperbolic maps
• Lyapunov exponents
• orbit determination

When dealing with an orbit determination problem, uncertainties arise from intrinsic errors related to observation devices and approximation models, and can be geometrically summarized in confidence ellipsoids. Several celestial mechanics problems deal with objects orbiting in ordered or chaotic regions of space. It immediately becomes significant being able to perform accurate orbit determination processes, especially when working in a chaotic environment, where real orbits in the long term might drastically diverge from the previously computed ones. We treat the problem from a theoretical point of view, modelling the orbit determination processes on discrete dynamical systems. We investigate the asymptotic behaviour of the confidence ellipsoids while the number of observations and the timespan over which they are performed increase. We employ ergodic and linear algebra arguments to describe the dimensions of the ellipsoids, confirming some numerical evidences: regular dynamics produces a polynomial decay of the uncertainties, while chaos gives an exponential rate for the initial conditions alone and a strictly-slower-than-exponential speed if an extra parameter is included. We improve some of these facts, providing conditions that imply a non-faster-than-polynomial rate of decay in the chaotic case with the parameter. We also apply these findings to well known examples of chaotic maps.