Hybrid dynamics of H\'enon mappings

TL;DR

This paper explores the degeneration of Hénon map dynamics using hybrid space theory, showing how invariant measures and Lyapunov exponents behave in the limit as parameters approach a singularity.

Contribution

It applies hybrid space theory to Hénon maps, analyzing the limits of invariant measures and Lyapunov exponents during degenerations.

Findings

Invariant measures converge to a measure on the Berkovich affine plane.

Lyapunov exponents have a well-defined limit in the degeneration process.

The approach links complex and non-archimedean dynamics through hybrid spaces.

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.