A structure theorem for semi-parabolic Hénon maps

2014/11/14 by Remus Radu, Radu, Remus, Raluca Tanase +1 · 2 citations
Mathematics · Physics and Astronomy · #37F20 #37F45 #47H10 #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1411.3824

openalex publication_date 2014/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider the parameter space P_λ⊂ ℂ² of complex Hénon maps H_c,a(x,y)=(x²+c+ay,ax), a≠ 0 which have a semi-parabolic fixed point with one eigenvalue λ=e^{2πi p/q}. We give a characterization of those Hénon maps from the curve P_λ that are small perturbations of a quadratic polynomial p with a parabolic fixed point of multiplier λ. We prove that there is an open disk of parameters in P_λ for which the semi-parabolic Hénon map has connected Julia set J and is structurally stable on J and J⁺. The Julia set J⁺ has a nice local description: inside a bidisk \mathbbD_r× \mathbbD_r it is a trivial fiber bundle over J_p, the Julia set of the polynomial p, with fibers biholomorphic to \mathbbD_r. The Julia set J is homeomorphic to a quotiented solenoid.

Cited by

• Structural Instability of Semi-Siegel Hénon maps
• Semi-parabolic tools for hyperbolic Hénon maps and continuity of Julia sets in ℂ²

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