Interesting Dynamics and Inverse Limits in a Family of One-Dimensional Maps

1. INTRODUCTION. In the past thirty or forty years inverse limits have been used extensively in dynamical systems as well as in continuum theory as a means to attack a myriad of unsolved problems. A study of one-dimensional branched manifolds by R. F. Williams [16] provided an early demonstration of the utility of the inverse limit construction in dynamical systems. In continuum theory, many surprising and complicated examples have been constructed using inverse limits. For example, a continuum admitting only constant maps and the identity as continuous transformations of the continuum into itself was constructed by Howard Cook [4] in the 1960s via inverse limits. The inverse limit technique is particularly useful in that it allows one to build complicated structures out of simpler ones. For instance, in Example 2 of this paper (see Figure 2) we demonstrate how a continuum containing a sin(1/x)-curve results from an inverse limit on the interval [0, 1] using a single simple piecewise-linear continuous function on the interval. Example 3 (see Figure 6) shows that even more complicated continua (indecomposable continua) may show up in inverse limits on intervals with equally simple looking bonding maps. This particular example actually consists of two copies of a continuum naturally associated with the " Smale horseshoe " glued together at a common end-point. Interest in inverse limits is increased by the realization that through the inverse limit construction one is able to turn the study of a dynamical system consisting of a space and a continuous function on that space into the study of another space with a homeomorphism on that space (Theorem 3.9). After reading with interest a paper by Susan Bassein that appeared in this MONTHLY in 1998 [2], we decided that the simple two-parameter family of