2025/02/26 by Sayani Bera, Bera, Sayani
Mathematics · #32M18 #32Q02 #32T05 #37F80 #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2502.19358
openalex publication_date 2025/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let H be a Hénon map of the form H(x,y)=(y,p(y)-ax). We prove that the escaping set U+ (or equivalently, the non-escaping set K+), of H is rigid under the actions of automorphisms of ℂ2 if the degree of H=d≤ |a|. Specifically, every automorphism of ℂ2 that preserves U+, essentially takes the form C ∘ Hs where s ∈ ℤ, and C(x,y)=(ηx, ηd y) with η some (d2-1)-root of unity. Consequently, we show that the automorphisms of the short ℂ2's, obtained as the sub-level sets of the (positive) Green's function corresponding to the Hénon map H for strictly positive values, are essentially linear maps of ℂ2 preserving the escaping set U+. Hence, the automorphism groups of these short ℂ2's are the same, finite, and form a subgroup of ℤd2-1.