C-19 - Topological Entropy

Abstract

It is important to be able to measure how complicated the dynamics of ƒ is, how many very different orbits it has, and how fast it mixes together various sets, etc. To some extent, this can be measured by topological entropy. In general, the space need not be metric, the iterates of one map may be replaced by an action of a fairly general semi group or group (for instance, ℤn or ℝn), etc. However, the basic ideas are similar in all cases. The notion of topological entropy is analogous to the notion of metric entropy; it is also called measure entropy or measure theoretical entropy. An important class of sub shifts are the sub shifts of finite type, also called topological Markov chains. Such a sub shift is determined by an s X s matrix M. Important examples of maps with positive entropy are those with horseshoes. The classical small horseshoe can be represented as a stretched square ABCD, bent as a horse shoe and reinserted. This construction is local; it is possible to extend this map to a homeomorphism of a larger compact set into itself.