Equidistribution of saddle periodic points for Hénon-like maps

TL;DR

The paper proves that saddle periodic points of a Hénon-like map in any dimension equidistribute with respect to the equilibrium measure $\mu=T^+\wedge T^-$ under the natural dynamical-degree condition $d>d_{p-1}^+$ and $d>d_{k-p-1}^-$. The authors develop a non-compact pluripotential framework built from Green currents, woven/tame current theory, and a coherent intersection theory via super-potentials and density theory, augmented by Oseledec measures and Pesin sets. The core strategy relates saddle points to intersections of the graph of $f^n$ with the diagonal in $D\times D$, then analyzes tangent currents after lifting to Grassmannians and employing ramified-covering arguments near the diagonal to control tangencies; this culminates in the convergence $d^{-n}\sum_{z\in Q_n}\delta_z\to \mu$ for saddle points $Q_n$ of period $n$. The work generalizes prior 2D and higher-dimensional results, providing a robust, scalable method for equidistribution in non-compact settings and advancing the intersection theory for positive closed currents in several complex variables.

Abstract

We prove that under the natural assumption over the dynamical degrees, the saddle periodic points of a Hénon-like map in any dimension equidistribute with respect to the equilibrium measure. Our work is a generalization of results of Bedford-Lyubich-Smillie, Dujardin and Dinh-Sibony along with improvements of their techniques. We also investigate some fine properties of Green currents associated with the map. On the pluripotential-theory side, in our non-compact setting, the wedge product of two positive closed currents of complementary bi-degrees can be defined using super-potentials and the density theory. We prove that these two definitions are coherent.

Theorems & Definitions (94)

• Theorem 1.1
• Definition 2.1
• Example 2.2
• Lemma 2.3
• proof
• Definition 2.4
• Remark 2.5
• Example 2.6
• Proposition 2.7
• proof