Appears in collection : Summer School 2025 - Cremona Group

A fundamental invariant for the dynamics of a Cremona transformation f:Pn?Pn is its "first dynamical degree"\, an asymptotic measure of how quickly the degree of a hypersurface grows when pulled back by the iterates of f. In many particular situations first dynamical degrees can be realized and effectively computed as eigenvalues of integer matrices. When the dimension n is two\, this is always the case. I will discuss a joint work with Jason Bell\, Mattias Jonsson and Holly Krieger in which we construct examples of Cremona transformations in dimensions three and higher with first dynamical degrees that are transcendental numbers. Analyzing the examples involves\, among other things\, interesting toric geometry and substantial results about diophantine approximation.