Conjugacy of Maps

Abstract

The notion of topological conjugacy is very important in connecting the dynamics of different maps. It relates the properties among maps via conjugacy relations. In other way, conjugacy is a change of variables that transforms one map into another. The variable changes from one system to another, that is, transition mappings are invertible and continuous, but not necessarily affine. A map $T:X \to X$ is an affine transformation, if for any $s,t \in X$ and any $\alpha \in \left[ {0,1} \right]$ such that $T\left( {\alpha s + \left( {1 - \alpha } \right)t} \right) = \alpha T\left( s \right) + \left( {1 - \alpha } \right)T\left( t \right)$. In general, it has three notions, viz., (i) shrink, (ii) rotate and (iii) translate the object. Two maps are said to be conjugate if they are equivalent to each other and their dynamics are similar. Specifically, two topologically conjugate maps share almost any dynamical property which does not explicitly require differentiation. Further it carries many properties of chaotic dynamics. Conjugacy is an equivalence relation among maps. In conjugacy relation, the transformation should be a homeomorphism (1–1, onto, bi-continuous), so that some topological structures are preserved. Naturally, it is a useful and also a wise trick to find conjugacy between a map and a simple map. Besides conjugacy, the concept of semi-conjugacy is also useful in many contexts with some limitations. It is not always possible to establish conjugacy among maps but may be possible to find semi-conjugacy relation among them. Semi-conjugacy is a less stringent relationship among maps than a conjugacy relationship. In this chapter some definitions and important theorems on conjugacy and semi-conjugacy relations have been given and discussed elaborately with examples. The dynamics on circle, the notion of orientation-preserving and orientation-reversing homeomorphisms, rotation and lift functions and their properties are explored. At the end the famous Denjoy’s theorem in diffeomorphism with irrational rotation number and conjugacy is briefly discussed.