Hybrid dynamics of hyperbolic automorphisms of K3 surfaces

TL;DR

This paper investigates the behavior of invariant measures of hyperbolic automorphisms on degenerating families of K3 surfaces using hybrid space theory, establishing measure convergence in degenerations.

Contribution

It introduces a framework for analyzing measure degeneration of hyperbolic automorphisms on K3 surfaces via hybrid spaces, connecting complex and non-archimedean dynamics.

Findings

Invariant measures $\u03b7_t$ converge weakly to $b7_0$ as $t o 0$

Establishes a link between complex and non-archimedean dynamics for K3 automorphisms

Provides a new perspective on degenerating hyperbolic dynamics using hybrid space theory.

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.