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Published online by Cambridge University Press: 05 June 2012
In the previous chapter we introduced the idea of a mapping with sensitive dependence and mentioned the relevance of this idea to the study of chaotic behaviour.
For simplicity, emphasis was given to mappings showing sensitive dependence at a single point of their domains. In this chapter, however, the emphasis shifts to mappings which show sensitive dependence at every point in their domains.
Examples given in the last chapter suggest that sensitive dependence of a mapping occurs at a point x of its domain with the following property: graphs of higher iterates rise steeply over each small interval containing the point x. Hence for a mapping to be sensitive everywhere the graphs of higher iterates must rise steeply above each small interval in the domain.
In this chapter we prove that mappings with wiggly iterates have sensitive dependence everywhere. The higher iterates of these mappings wiggle up and down between 0 and 1. Above every small interval in the domain, we can find an iterate with slope as large as we please.
The wiggly behaviour of the iterates will be shown to imply that the mapping has not only
(a) sensitive dependence everywhere, but also
(b) transitivity, and
(c) a dense set of periodic points.
In a popular definition, due to the U.S. mathematician Robert Devaney, these three properties are taken as the essential ingredients of chaos.
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