Sampling Hyperspheres via Extreme Value Theory: Implications for Measuring Attractor Dimensions

The attractor Hausdorff dimension is an important quantity bridging information theory and dynamical systems, as it is related to the number of effective degrees of freedom of the underlying dynamical system. By using the link between extreme value theory and Poincare recurrences, it is possible to estimate this quantity from time series of high-dimensional systems without embedding the data. In general $$d \le n$$ , where n is the dimension of the full phase-space, as the dynamics freezes some of the available degrees of freedom. This is equivalent to constraining trajectories on a compact object in phase space, namely the attractor. Information theory shows that the equality $$d=n$$ holds for random systems. However, applying extreme value theory, we show that this result cannot be recovered and that $$d