Continuous itinerary functions and dendrite maps

We examine the “kneading sequence” theory of maps on dendrites, concentrating on those maps having one “turning point” and the “unique itinerary property” (i.e., distinct points have distinct itineraries). This theory has major overlaps with the theories of polynomial Julia Sets and Hubbard Trees, but also has significant differences from those two theories (for which the unique itinerary property does not always hold). We show that the unique itinerary property is a powerful property, and allows a simple classification of such dendrite maps with respect to their kneading sequences (up to conjugacy if there is no nontrivial invariant subdendrite).One of the major tools introduced here is the continuous itinerary function. If one takes the set of all sequences of the symbols used to define the itineraries with respect to a partition of a space, there is a natural topology which forces the itinerary function (from the original space into the space of sequences of symbols) to be continuous, although this often leads to a non-Hausdorff itinerary topology. Despite this apparent drawback, we show that this itinerary topology is a useful tool for analyzing the dynamics of continuous maps on metric spaces.Characterizations in terms of kneading sequences are given for topological properties of these maps, including various transitivity properties, (in)decomposability of inverse limits, and the existence of certain “piecewise linearizations” of such maps which are a natural generalization of “tent” maps on the interval. The itinerary topology provides a natural topology for the parameter space of all kneading sequences, a natural subspace of which will be shown to have a one-point compactification that is a dendrite, with an interesting connection to the Mandelbrot Set.