On Dynamics of Continuous Functions

Abstract

The discrete dynamical system of a continuous self-map is generated by iteration of the map; however, the iteration itself, being an operator on the space of continuous self-maps, may generate unusual dynamical behaviours. In this thesis, we prove that the iteration operator is continuous on the space of all continuous self-maps of a compact metric space and therefore induces a discrete dynamical system on the space. We also show how its fixed points and periodic points are determined, and characterize them in the case that the compact metric space is a compact interval or the unit circle by discussing the Babbage equation. Furthermore, we prove that all orbits of the iteration operator are bounded, but most fixed points are not stable. The boundedness and instability exhibit a complex behaviour of the iteration operation, but we prove that this complex behaviour is not chaotic in Devaney’s sense. Another complicated yet critical discrete dynamical system is that which emanates due to a continuous piecewise monotone self-map on an interval. In the kneading theory developed by Milnor and Thurston, it is proved that the kneading matrix and the kneading determinant associated with such a map are invariants under orientation-preserving conjugacy. We consider whether this result is valid for orientation-reversing conjugacy. We also present applications of obtained results towards the computational complexity of kneading matrices and the classification of maps up to topological conjugacy. Furthermore, a relation between kneading matrices of maps and their iterates for a class of chaotic maps is described. Closely related is the theory of iterative equations. There are obtained many results on solutions of such equations involving a linear combination of iterates, called polynomial-like iterative equations. We investigate an iterative equation with multiplication, a nonlinear combination of iterates, and give results on the existence, uniqueness, stability, and construction of its continuous solutions. Our study not only addresses essential problems in the theory of dynamical systems and iterative equations but also exhibits subtle interplay between these two areas.