Diophantine geometry and arithmetic dynamics

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My research in number theory deals largely with problems coming from dynamical systems. In particular, my research deals with questions of the following form: given a rational function and some initial value (a rational number), what are the arithmetic properties of the sequence of numbers generated by repeatedly applying the function to the initial value. For example, if a sequence of this form falls into a repeating cycle, one would like to know how long such a cycle might be, given some information about the function. One the other hand, if the sequence never repeats, then the "heights" of the terms (essentially the number of digits in numerator and denominator of these rational numbers) grow at an exponential rate. One significant problem in the area, one which I have made some progress on, and plan to research further, is to obtain explicit lower bounds on the exact rate of this exponential growth, for certain families of rational functions. This research has implications in various areas of number theory (to elliptic curves, recurrence sequences, etc.), as well as to problems in dynamics, computation, and mathematical logic.