Further results on expansive mappings

Abstract

In this paper, several theorems are proved concerning the concepts of expansiveness and asymptoticity from topological dynamics. The results are derived using the techniques the author developed in a previous paper in this journal. In [8], this author showed how the concept of an expansive homeomorphism could be generalized to that of an expansive continuous relation (called an expansive mapping), and that several of the well-known theorems on expansive homeomorphisms generalized in this new setting. One reason expansive mappings are important is that they furnish the only routine technique (by using the shift transformation) for producing expansive homeomorphisms. One surprising result the technique has yielded is a techhomologically trivial continuum supporting an expansive homeomorphism. (See [8].) It is useful to know how expansive mappings resemble expansive homeomorphisms so that one might gain some insight as to what surprises this technique might not produce. In this paper, several more well-known theorems will be generalized by using the tools developed in [8]. For reference purposes, the basic definitions and techniques from [8] will now be given. If X is a metric space with metric d, and if f is a homeomorphism of X onto itself, then f is said to be expansive on X with expansive constant 6 > 0 if x, y E X, x # y, implies d(f '(x), f f(y)) > 8 for some integer n. Distinct points x and y are said to be positively (negatively) asymptotic under f if for each E > 0, there is an integer N such that n > N (n 0 such that d(x,y) < -q implies f(y) C N.) Henceforth, a continuous multivalued transformation will simply be called a mapping. DEFINITION 1. Let x E X. The orbit of x under f is defined by O(x) = UOO Of n (x). DEFINITION 2. Let x E X. A suborbit of x under f is a set of the form Received by the editors October 18, 1975 and, in revised form, December 15, 1975. AMS (MOS) subject classifications (1970). Primary 54H20; Secondary 54C60, 54C65.