Hybrid dynamics of H\'enon mappings

Abstract

For studying the meromorphic degeneration of complex dynamics, the theory of hybrid spaces, introduced by Boucksom, Favre and Jonsson, is known to be a strong tool. In this paper, we apply this theory to the dynamics of H\'enon maps. For a family of H\'enon maps $\{H_t\}_{t\in\mathbb{D}^*}$ that is parametrized by a unit punctured disk and meromorphically degenerates at the origin, we show that as $t\to 0$, the family of the invariant measures $\{\mu_t\}$ "weakly converges" to a measure on the Berkovich affine plane associated to the non-archimedean H\'enon map determined by the family $\{H_t\}_t$. We also calculate the limit of their Lyapunov exponents.