Unbounded itineraries for the exponential family

Consider the complex exponential family E_λ (z) = λez, λ ∈ $C$. Using symbolic dynamics Devaney and Krych were able to study the dynamics of these maps by assigning infinite sequences of integers (or itineraries) to each orbit. Not every sequence corresponds to an orbit of E_λ; those sequences which grow too fast must be eliminated. However, given any ‘allowable’ sequence, Devaney and Krych showed that there are infinitely many complex numbers with that itinerary. Furthermore, in an effort to study the set of points which share the same itinerary, they showed that for any given regular and bounded sequence, this set is nothing but a hair, i.e., a continuous curve that extends to ∞ in the right-hand half plane and which accumulates on a single point, usually called the endpoint (or the landing point) of the hair. For each such hair, the orbit of the endpoint remains bounded, while all other points on the hair escape to ∞ under iteration.; In this paper we study those itineraries which are regular, but unbounded. We will show that, as in the Devaney and Krych's case, the set of points which share one such itinerary forms also a hair. Unlike the bounded case however, we will show that while all the points on this hair escape to ∞ under iteration, the endpoint does so at a much slower rate than all the rest. In the subsequent chapters we will explore in greater detail some other properties of the endpoint which we know hold true in the case of bounded itineraries. These properties give another way of distinguishing between the endpoint of the hair and all the other points on it.; We conclude the thesis with some examples which are meant to illustrate how the algorithm works in concrete and which helped us form a better insight to the more general problem.