Given a complex variety \(X\) and a rational self-map \(f: X\to X\), one of the most important quantities in understanding the corresponding dynamical system is the dynamical degree, which is a nonnegative real number that gives some measure of how complex the system is. The dynamical degree is often a nonnegative integer and in many settings has been proved to be an algebraic number, including the case of endomorphisms of the plane. This naturally leads to the question: can the dynamical be transcendental? We show that the answer to this question is `yes’, by giving a rational self-map \(f\) of the projective plane and using techniques from Diophantine approximation to exhibit the transcendence of the dynamical degree \(f\).

This is joint work with Jeff Diller and Mattias Jonsson.