Dynamical Systems

1. What You Need to Know to Knead

Milnor and Thurston have developed a so-called Kneading Theory to analyse the iterates of piecewise monotone mappings on an interval.
This first appeared as a preprint in 1977 and in published form in 1988 as On iterated maps of the interval (in Springer Lecture Notes in Mathematics, Vol. 1342).

In 1989 I wrote a paper titled What You Need to Know to Knead (Advances in Mathematics, 78, 192-252) which presented a purely combinatorial approach to this theory. The above is a somewhat revised version of part of the paper.

In 1966 Parry showed that a topologically transitive continuous piecewise monotone mapping f with positive entropy h(f)) is conjugate to a uniformly piecewise linear mapping with slope exp(h(f)). In the following note Parry's result is generalised to what we call the class of essentially transitive mappings. This generalisation is of some interest in as much as for mappings with one turning point the converse also holds.

2. A Note on a Theorem of Parry (2010)

The following note presents an approach to studying the iterates of a mapping whose restriction to the complement of a finite set is continuous and open.The main examples to which the approach can be applied are piecewise monotone mappings defined on an interval or a finite graph.

3. Iterates of mappings which are almost continuous and open (2010)

The aim of this note is to bring attention to a class of discrete dynamical systems exhibiting some complex behavior. Each of these is defined as a self-mapping of the unit square and is obtained by coupling two families of self-mappings of the unit interval. I have provided a JavaScript program, accessible at www.uni-bielefeld.de/~preston/iterates.html, which can be used to 'discover' more about these mappings

4. Coupled one-dimensional dynamical systems(2012)