Hybrid dynamics of hyperbolic automorphisms of K3 surfaces

Abstract

We study degenerating families of hyperbolic dynamics over complex K3 surfaces by means of the theory of hybrid spaces by Boucksom, Favre, and Jonsson. For an analytic family of hyperbolic automorphisms $\{f_t: X_t\to X_t\}_{t\in\mathbb{D}^*}$ over K3 surfaces $X_t$ that is possibly meromorphically degenerating at the origin, we consider the family of invariant measures $\{\eta_t\}$ on $X_t$ constructed by Cantat. The family $f_t$ induces a hyperbolic automorphism $f_{\mathbb{C}((t))}^{\mathop{\mathrm{an}}}:X_{\mathbb{C}((t))}^{\mathop{\mathrm{an}}}\to X_{\mathbb{C}((t))}^{\mathop{\mathrm{an}}}$ over the induced non-archimedean K3 surface, where we also have a measure $\eta_0$ by Filip. Our main theorem states the weak convergence of $\{\eta_t\}$ to $\eta_0$ as $t\to0$ over the induced so-called hybrid space.