Dynamical degrees of birational transformations of projective surfaces

The dynamical degree λ ( f ) λ (f) of a birational transformation f f measures the exponential growth rate of the degree of the formulas that define the n n th iterate of f f . We study the set of all dynamical degrees of all birational transformations of projective surfaces, and the relationship between the value of λ ( f ) λ (f) and the structure of the conjugacy class of f f . For instance, the set of all dynamical degrees of birational transformations of the complex projective plane is a closed and well ordered set of algebraic numbers.