ABSTRACT
In this chapter we introduce the concepts of Dynamical Systems through iteration of functions of a single real variable. The main idea is to understand what the orbit of a point is like when iterated repeatedly by the same function: x1 = f(x0) and xn = f(xn Š 1) for n 1. In one dimension, the graph of the function can be used quite easily to analyze the iterates of a point by a function. Also, because of the restriction to one dimension, the concepts of a periodic orbit being attracting or repelling become a simple consequence of the Mean Value Theorem. Two other important ideas in Dynamical Systems are the notion of topological conjugacy and symbolic dynamics. Two functions f and g are said to be topologically conjugate if there is a homeomorphism h with g(x) = h f h Š 1(x). In one dimension, we are able to prove quite easily that simple examples such as f(x) = 2x and g(x) = 3x are topologically conjugate. For the quadratic family, F
(x) = µx(1 Š x), we can show that if µ, µ > 4, then F µ
and F µ
are topologically conjugate even though both maps have infinitely many periodic points. We also use the quadratic map to introduce the ideas of symbolic dynamics. The idea is that we label two intervals I1 and I2 and are able to show that sequences of these two intervals are in one-to-one correspondence with orbits of the map under iteration. The final section of the chapter concerns iteration of homeomorphisms of the circle. In this setting, the average rotation of a point under repeated iteration, the rotation number, determines whether the map has periodic points or not.
