Monotonicity of dynamical degrees for H{\'e}non-like and polynomial-like maps

TL;DR

This paper proves that for H{é}non-like and polynomial-like maps in any dimension, the sequence of dynamical degrees exhibits a monotonic pattern, increasing up to a maximum and then decreasing, a property previously unproven outside algebraic contexts.

Contribution

It establishes the monotonicity of dynamical degrees for non-algebraic H{é}non-like and polynomial-like maps in any dimension, using pluripotential theory and deformation of positive closed currents.

Findings

Dynamical degrees form an increasing-then-decreasing sequence.

First proof of this property outside algebraic setting.

Utilizes deformation of currents and pluripotential theory.

Abstract

We prove that, for every invertible horizontal-like map (i.e., H{\'e}non-like map) in any dimension, the sequence of the dynamical degrees is increasing until that of maximal value, which is the main dynamical degree, and decreasing after that. Similarly, for polynomial-like maps in any dimension, the sequence of dynamical degrees is increasing until the last one, which is the topological degree. This is the first time that such a property is proved outside of the algebraic setting. Our proof is based on the construction of a suitable deformation for positive closed currents, which relies on tools from pluripotential theory and the solution of the $d$, $\bar \partial$, and $dd^c$ equations on convex domains.