Measuring Complexity Growth in Dynamical Systems
Joseph H. Silverman (Brown University)

Dynamics is the study of iteration of a function f : X –> X. A coarse measure of the complexity of f is its degree. The average degree of the iterates of f is called the *dynamical degree* of f. Formally,
DynDeg(f) = lim (deg f^n)^{1/n}.
Similarly, a coarse measure of the complexity of an arithmetic object, such as an integer, is its height, which is the number of bits required to store the object on a computer. The *arithmetic degree* of a point x in X is the average height of the points in its orbit. Formally,
ArithDeg(f,x) = lim (height f^n(x))^{1/n}.
In this talk I will discuss dynamical and arithmetic degrees and a fundamental conjecture that relates them, and, as time permits, describe recent work on the variation of dynamical degrees in families.