On automorphisms of blowups of ${\mathbb{P}}^3$

Let $\pi :X\to {\mathbb{P}}^3$ be a finite composition of blowups along smooth centers. We show that for “almost all” of such $X$, if $f\in Aut(X)$ then its first and second dynamical degrees are the same. We also construct many examples of finite blowups $X\to {\mathbb{P}}^3$, whose automorphism group $Aut(X)$ has only finitely many connected components. We also present a heuristic argument showing that for a “generic” compact K"ahler manifold $X$ of dimension $\ge 3$, the automorphism group $Aut(X)$ has only finitely many connected components.