Robert M. Corless
Tue Nov 28 18:16:03 PST 1995
In ``Continued Fractions and Chaos'' I used, without explanation, the following definition of S.I.C. or Sensitive to Initial Conditions. It seemed to me to be obviously equivalent. On thinking about it a bit more, it's not so obvious, but it is indeed a useful way of looking at it. For example, unlike the Lyapunov exponent definition, it does not require any smoothness of f at all. Besides, I have not seen this definition elsewhere in the literature and so it might be as well to write it down explicitly.
We suppose that we are iterating a map here, with orbit where as usual.
Definition. A map f is S.I.C. if for almost all in the domain of f there exist constants , and such that for all there exist (the ball of radius around ) such that for some we have .
Roughly speaking, , where is the Lyapunov exponent, and if we have exponential separation of orbits, then , (at least until the nonlinearities and boundedness takes over---a better formulation of Lyapunov exponent looks directly at the tangent space, whereas we are looking here at the actual orbits) which is the usual definition of S.I.C. if . If we plug in the k from the above definition we get which, if we take , gives us the required behaviour.
In shorthand notation we say that points initially apart become apart after iterations of the map. In some ways this is an easier definition to work with than the usual one, which requires investigations of limits in tangent space (i.e. calculation of the Lyapunov exponent), which may or may not even exist for a large number of orbits. In my view it is superior to some other definitions of S.I.C. (such as Wiggins' definition) which do not require separation at an exponential rate. To me, as it was to Paddy Nerenberg, it is the exponential rate that makes a map `sensitive', because algebraic separation is usually by far the less spectacular. However, I acknowledge that so long as we are clear about which definition is being used, different definitions can be used in different contexts, as appropriate.
What we proved in the paper was that we could take M from the known speed of the Euclidean algorithm, and choose . In fact, can be explicitly inferred from the minimal Lyapunov exponent theorem of the paper. So for the Gauss map, the results are uniform, in some sense. Indeed, we can also take , the width of the entire interval.
For the shift map mod 1, this definition is also easy to use, and I recommend the reader show that this map is S.I.C. as an exercise.