Introduction

Continued Fractions and Chaos ~~~~~~ Robert M. Corless

The theory of continued fractions goes back at least to c. A. D. 500 to the work of ryabhata, and possibly as far back as c. 300 B.C. to Euclid. The theory of chaotic dynamical systems is relatively recent, going back only to the work of Poincaré [22] and Birkhoff [2]. The foundations of the theory of continued fractions, as we know it now, are well established due to the work of Euler, Lagrange, Gauss, and others, while the foundations of chaotic dynamical systems are still evolving. This paper will use the well-established theory of simple continued fractions to explore some current results of the theory of chaotic dynamical systems.

Olds [20] gives a good introduction to the classical theory of simple continued fractions, by which we mean continued fractions of the form

Simple continued fractions have found applications in Fabry-Perot interferometry [13], and in the concept of noble numbers used in orbital stability and quasi-amorphous states of matter [25]. For other uses of simple continued fractions in chaos, see [8]. Other types of continued fraction exist, for example, Gautschi [9], Henrici [12], Jones and Thron [14], and others, use functional or analytic continued fractions in approximation theory, since analytic continued fractions can be very effective for computation. We will not be concerned with such continued fractions. We will summarize in the next section all the classical results that we need, without proof. Proofs can be found in [20,11,19,15,1], and [18].