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

Bianchi, Fabrizio; Dinh, Tien-Cuong; Rakhimov, Karim

doi:10.1090/tran/9225

Monotonicity of dynamical degrees for Hénon-like and polynomial-like maps
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by Fabrizio Bianchi, Tien-Cuong Dinh and Karim Rakhimov;

Trans. Amer. Math. Soc. 377 (2024), 6595-6618

DOI: https://doi.org/10.1090/tran/9225

Published electronically: June 28, 2024

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Abstract:

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.

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