Geometry of repeated measurements in chaotic systems.

We use joint probability matrices for measurements at different times to describe chaotic systems. By coarse graining the range of the measured variable into uniformly sized bins we can generate matrices that contain both topological and metric information about the systems being studied. Armed with this tool we examine two extreme families of chaotic systems. In the case of one-dimensional piecewise linear maps, we can construct transfer matrices that depend on the map and partition used, and which allow us to generate the respective joint probability matrices for all times as well as the exact time evolution of the mutual information function. We find that the mutual information decays linearly or exponentially depending on whether the second-largest eigenvalue of the transfer matrix is zero or not. In the case of three-dimensional, continuous-time chaotic systems we generate the joint probability matrices directly from numerical data. We show that these matrices directly provide attractor reconstructions with information about the attractor's probability measure.