Monotonicity of dynamical degrees for Hénon-like and polynomial-like maps

Fabrizio Bianchi (1, 2) , Tien-Cuong Dinh (3) , Karim Rakhimov (3)

1 CNRS - Centre National de la Recherche Scientifique
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 NUS - National University of Singapore

We prove that, for every invertible horizontal-like map (i.e., Hénon-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.

Mots clés: Horizontal-like maps; Polynomial-like maps; Dynamical degrees; Deformation of currents

Domaines: Systèmes dynamiques [math.DS]; Variables complexes [math.CV]

Fabrizio Bianchi, Tien-Cuong Dinh, Karim Rakhimov. Monotonicity of dynamical degrees for Hénon-like and polynomial-like maps. 2023. ⟨hal-04163951⟩