A transcendental birational dynamical degree

The dynamical degree of an invertible self-map of projective space is an asymptotic measure of the algebraic complexity of the iterates of the map. This numerical invariant controls many aspects of the dynamics of the map, and in this talk I will survey the significance of the dynamical degree and discuss some important examples. In these examples, the dynamical degree is an integer or an eigenvalue of an integer matrix, so an algebraic number, as was conjecturally the case for all such maps. I will discuss joint work with Bell, Diller, and Jonsson in which we refute this conjecture by constructing invertible maps of projective 3-space which have transcendental dynamical degree.