H\"older continuity and laminarity of the Green currents for H\'enon-like maps

TL;DR

This paper proves that Green currents for Hénon-like maps exhibit Hölder continuity and laminarity, extending known results from algebraic cases and two-dimensional settings to higher dimensions.

Contribution

It establishes Hölder continuity of super-potentials and laminarity of Green currents for Hénon-like maps in any dimension under natural assumptions.

Findings

Green currents have Hölder continuous super-potentials.

The measure of maximal entropy is a Monge-Ampère of a Hölder continuous plurisubharmonic function.

Green currents are woven and laminar when of bidegree (1,1).

Abstract

Under a natural assumption on the dynamical degrees, we prove that the Green currents associated to any H\'enon-like map in any dimension have H\"older continuous super-potentials, i.e., give H\"older continuous linear functionals on suitable spaces of forms and currents. As a consequence, the unique measure of maximal entropy is the Monge-Amp\`ere of a H\"older continuous plurisubharmonic function and has strictly positive Hausdorff dimension. Under the same assumptions, we also prove that the Green currents are woven. When they are of bidegree $(1,1)$, they are laminar. In particular, our results generalize results known until now only in algebraic settings, or in dimension 2.