Equidistribution of saddle periodic points for H\'enon-like maps

TL;DR

This paper proves that saddle periodic points of Hénon-like maps in any dimension become uniformly distributed according to the equilibrium measure, extending and improving previous results in complex dynamics.

Contribution

It generalizes existing equidistribution results for Hénon-like maps to higher dimensions and introduces refined techniques involving Green currents and super-potentials.

Findings

Saddle periodic points equidistribute with respect to the equilibrium measure.

The wedge product of positive closed currents is well-defined and consistent in the non-compact setting.

The work extends complex dynamics techniques to higher dimensions with new insights into Green currents.

Abstract

We prove that under the natural assumption over the dynamical degrees, the saddle periodic points of a H\'enon-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.