Dynamical degrees of birational transformations of projective surfaces

The dynamical degree lambda( f )  of a birational transformation f measures the exponential growth rate of the degree of the formulae that define the n -th iterate of f  . We study the set of all dynamical degrees of all birational transformations of projective surfaces, and the relationship between the value of lambda( f ) and the structure of the conjugacy class of 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.