Proceedings of Symposia in Applied Mathematics Volume 39, 1989 Dynamics of Simple Maps ROBERT L. DEVANEY Our goal in this paper is to introduce, in as simple a setting as possible, some of the fundamental ideas of dynamics. These ideas include the notions of chaos and fractal, two often used and abused words in contemporary math- ematical jargon. Our aim is to show how these terms arise naturally in the field of dynamical systems. One of the great breakthroughs in dynamics in the past few decades has been the realization that even the simplest of dynamical systems may behave extremely unpredictably. The quadratic map of the real line given by x —• x2 + c where c is a real parameter is typical. Certainly there is no simpler nonlinear dynamical process than iteration of this function. Yet, as we will see, there are many ovalues where the dynamics of this map are extremely complicated. When we pass to the complex plane, the mapping z z2 + c is the setting for the beautiful recent mathematical work of Mandelbrot and Douady and Hubbard. For these maps, the plane decomposes into two distinct subsets, the stable set, on which the dynamics are tame and well understood, and the Julia set, on which the dynamics are quite chaotic and complicated (yet, thanks to recent work, fairly well understood). The contributions of Keen and Branner in this volume delve in more detail into these issues. The place where one finds complicated dynamics is often a fractal. This is almost always true in the case of Julia sets, and in many other settings as well, including the basin boundaries studied by Yorke and Alligood and the "horseshoe" mappings described by Holmes in this volume. More details on fractals and some of their applications are described by Harrison and Barnsley in this volume as well. Our goal in this paper is to lay the foundation for these later contributions. We will introduce a number of fundamental dynamical notions in the setting of one-dimensional dynamics. These notions include the notions of periodic 1980 Mathematics Subject Classification (1985 Revision). Primary 58F13 Secondary 58F14, 30C99. ©1989 American Mathematical Society 0160-7634/89 $1.00 + $.25 per page 1
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