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  and Birkhoff . 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  gives a good introduction to the classical theory of simple continued fractions, by which we mean continued fractions of the formwhere the are all positive integers, except which may be zero or negative. We will denote this as , and in what follows will usually be zero.
Simple continued fractions have found applications in Fabry-Perot
interferometry , and in the concept of noble
used in orbital stability and quasi-amorphous states of
For other uses of simple continued fractions in
chaos, see .
Other types of continued fraction exist, for
example, Gautschi , Henrici ,
Jones and Thron , 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],