Aspects of universality in function iteration.

摘要

This work deals with some aspects of universal topological and metric dynamic behavior of iterated maps of the interval. It is divided into three chapters.;Chapter two explains some universal topological dynamic behavior first observed via computer experimentation in 1973. The explanation reveals the intimate relationship between the root structure of a one parameter family of maps and its collection of kneading sequences. An explicit algorithm is given to construct all kneading sequences of arbitrarily long but finite length.;Chapter three addresses the problem of hypotheses on collections of one parameter families sufficient to ensure a certain strong type of common topological dynamic behavior. In particular, a counterexample is constructed to a published conjecture on such sufficient conditions.;Chapter one introduces important basic concepts and ends with a discussion of the connection equation for one parameter families of maps of the interval. This equation describes the local behavior of forward and pre-orbital points near parameter values corresponding to points of super stability. It is here that combinatorial aspects of the kneading sequence are seen to relate to the dynamics of the map as the parameter varies.