Non-archimedean and hybrid dynamics of regular polynomial automorphisms

In this paper, we study the weak limit of the invariant measures attached to an analytic family $\{f_t{\}}_{t\in {\mathbb{D}}^{\ast }}$ of regular polynomial automorphisms over ${\mathbb{C}}^N$ parametrized by the unit punctured disk which may degenerate at the origin, by using the techniques called hybrid spaces developed by Boucksom–Favre–Jonsson. For each $t$, the invariant measure ${\mu }_t$ is constructed by Sibony, and we show that the weak limit of $\{{\mu }_t{\}}_t$ over the hybrid space is given by the invariant measure of the regular polynomial automorphism $f_{\mathbb{C}((t))}^{\mathrm{an}}$ over $\mathbb{C}((t))$ induced from the original family. This is a generalization of the author’s previous result for degenerating families of dynamics of Hénon mappings.