Two views on entropy in dynamical systems

Abstract

Banach's fixed point theorem is a part of standard curriculum of several university courses. It is also an example of a discrete dynamical system that is very regular -- in the limit, the orbit of each point ``ends'' at a single fixed point. This is the starting point for this article. We begin by analyzing how small changes in the assumptions of this theorem affect the regularity of the system. We then discuss how the concept of regularity and chaos can be formalized. With this goal in mind, we talk about topological entropy. We give definitions and some examples of topological and polynomial entropy in dynamical systems. We also explain two ways of looking at these dynamical invariants. We also consider points that are in a sense the opposite to fixed points, namely wandering points and at the end we explain the role of wandering points in measuring the complexity of a dynamical system.