## F. Bianchi - T. C. Dinh - K. Rakhimov

# Monotonicity of dynamical degrees for H{é}non-like and polynomial-like maps

created by bianchi on 29 Mar 2024

modified on 04 Jul 2024

[

BibTeX]

*Accepted Paper*

**Inserted:** 29 mar 2024

**Last Updated:** 4 jul 2024

**Journal:** Transactions of the AMS

**Year:** 2023

**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.