12 - Conjugacy

Summary

In the previous chapter we described some simple changes of coordinates on the line in terms of invertible affine mappings. These mappings are called transition mappings from one coordinate system to the other.

In this chapter we consider changes of coordinates in which the transition mappings are not necessarily affine, but are still invertible and continuous.

Topological conjugacy (or just conjugacy for short) is the relationship which exists between two mappings when one is obtained from the other by a change of coordinates. We shall prove that when two mappings are in this relationship, they have similar dynamics; in particular when one is chaotic so is the other.

Unfortunately, there is no systematic method for showing conjugacy between a given pair of mappings and it is usually difficult to prove it. We can prove, however, that the tent mapping T₄ and the logistic mapping Q₄ are conjugate. More generally, we prove that all maps with wiggly iterates are conjugate to the tent map.

DEFINITION AND EXAMPLES

This section is about changes of coordinates between two mappings which are no longer assumed to be affine. A transition mapping for any change of coordinates needs to be both continuous and invertible. Continuity ensures that points which are close together in one coordinate system will also be close in the other coordinate system. Invertibility ensures that we can ‘undo’ the change of coordinate.