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

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