Dynamical degrees of endomorphisms of affine surfaces}

Let $f: \mathbf C^2 \rightarrow \mathbf C^2$ be a polynomial transformation. The dynamical degree of $f$ is defined as $\lim_n (\text{deg} f^n)^{1/n}$, where $\text{deg} f^n$ is the degree of the $n$-th iterate of $f$. In 2007, Favre and Jonsson showed that the dynamical degree of any polynomial endomorphism of $\mathbf C^2$ is an algebraic integer of degree $\leq 2$. For any affine surface, there is a definition of the dynamical degree that generalizes the one on the affine plane. We show that the result still holds in this case: the dynamical degree of an endomorphism of any complex affine surface is an algebraic integer of degree $\leq 2$. In this talk, I will give an overview of the recent results obtained on dynamical degrees on algebraic varieties and explain the key tools of the proof.