Topological Properties

Abstract

In any experiment, one has to use a measuring apparatus which naturally has only a finite precision. In this section, we take the first steps to formalize this. One formalization will be based on partitioning the phase space into pieces and considering that the experiment will only tell us in which piece of the partition (called an atom) a point currently is, but not where inside this piece it happens to be. This is analogous to classical coarse-graining in statistical mechanics. (Later, we will also encounter partitions with different weights on the pieces.) Thus, we know only a fuzzy approximation of the true orbit of the system. As we shall see, one of the miracles appearing in hyperbolic systems is that this information alone, when accumulated over long enough time, will in fact tell us many details about the orbit. In physical applications, one often can observe only one orbit and the information one obtains is considered to be typical of the whole system. We discuss this in more detail in Chap. 9.