How Analytic Functions Preserve Closeness

Topological dynamics is an area of topology in which one studies the geometric properties of certain groups of transformations which had their origins in classical dynamics and systems of differential equations. Henri Poincare is generally credited with originating the subject. He was probably the first to solve dynamics problems as problems in topology. Poincare's work was later abstracted, systematized, and carried on by G. D. Birkhoff. An area of current research in Topological Dynamics is the study of the somewhat elusive "expanisve property". Roughly speaking, the expansive property deals with the ability of functions to "spread out" points by means of repeated application of the functions, i.e., iterations. This concept will be precisely defined later. In this paper, various types of expansiveness will be defined, and it will then be shown that in most cases, analytic functions are not expansive. The results are both new and easy to prove, and the paper gives the reader with a limited background in topology an introduction to an area of current research. In addition the proofs are good exercises in complex analysis. If f is a homeomorphism of a metric space (X, d) onto itself, then f is said to be expansive if there exists 8 >0 such that x,y E (X, d), x t=y implies d(fn(x),fJ(y)) > 5 for some integer n. Iff is only assumed to map (X, d) continuously into itself, the expansive property can be adjusted by calling f positively expansive if the integer n is non-negative. The concept of expansiveness can also be weakened by requiring that for each x E (X, d), there is a S(x) >0 such that if y E (X, d), y =x, then d(fn(x), ff(y)) >5(x) for some integer n. In this case, f is called pointwise expansive. A similar modification allows one to weaken positively expansive to pointwise positively expansive. Clearly, "expansive" and "positively expansive" are stronger than "pointwise expansive" and "pointwise positively expansive", respectively, and for homeomorphisms onto, i.e. bijections, "positively expansive" is stronger than "expansive". We now illustrate these concepts with several examples. Let us take (X, d) to be the real line with the usual distance function in the following two examples. First, if f(x) = 2x, f is a homeomorphism of (X, d) onto itself, and since.fn(x) = 2nx, and d(f"(x),fn(y)) = 2 I x -ye, we see that f is both expansive and positively expansive. (Choose 8 to be any positive real number.) If f(x) = 2x for x> 0 and x/2 for x 0 such that x,y E (X, d), x t=y implies d(f(x),f(y)) > d for some f E F. A similar generalization allows one to define pointwise expansive families. Clearly "expansive family" implies "pointwise expansive family", and these two concepts are implied by their respective predecessors, since if f is either expansive or pointwise expansive, we can take F= { fn: n = 0, ? 1,...), and if f is either positively expansive or pointwise positively expansive, we can take F= { fn: n = 0, 1,.. . ). These definitions are easily understood by anyone who knows what a metric space is, but some very simple questions about these properties remain unanswered. For instance, it has been known for over 18 years (see [7] and [8]) that there does not exist an expansive homeomorphism on the closed unit disk, but the existence or non-existence of an expansive homeomorphism on the closed unit ball in three dimensions is still an open question.