5 - Asymptotic orbits II: differentiable mappings

Summary

The main idea in this chapter is very simple. For a mapping which is differentiable, the graph has a tangent at each point. Near the point of tangency the graph stays very close to the tangent. But the tangent is the graph of an affine mapping and so the dynamics of the differentiable map should be close to that of the affine mapping. We use this to predict the dynamics of a differentiable mapping near a fixed point.

We begin by using the idea of zooming to help us express the ideas of tangency and differentiability in terms of modern computer graphics. Graphs are then used to motivate the main theorem on the dynamics of a differentiable mapping near a fixed point. This leads us to the ideas of attracting, repelling and indifferent fixed points.

Finally, we use the results for dynamics of mappings near fixed points to study their dynamics near periodic points and orbits.

DIFFERENTIABLE MAPPINGS

In this section we show how tangents to curves can be obtained by zooming. We then show the relevance of this to the dynamics of differentiable mappings near their fixed points.

Tangents

The problem of finding a tangent at some point on a curve is the geometric motivation for the study of differential calculus. Beginners in the subject are assumed to have an intuitive understanding of the idea of a tangent.