2015/08/14 by Remus Radu, Radu, Remus, Raluca Tanase +1
Mathematics · Physics and Astronomy · #32A99 #37D99 #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.1508.03625
openalex publication_date 2015/08/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove some new continuity results for the Julia sets J and J+ of the complex Hénon map Hc,a(x,y)=(x2+c+ay, ax), where a and c are complex parameters. We look at the parameter space of dissipative Hénon maps which have a fixed point with one eigenvalue (1+t)λ, where λ is a root of unity and t is real and small in absolute value. These maps have a semi-parabolic fixed point when t is 0, and we use the techniques that we have developed in [RT] for the semi-parabolic case to describe nearby perturbations. We show that for small nonzero |t|, the Hénon map is hyperbolic and has connected Julia set. We prove that the Julia sets J and J+ depend continuously on the parameters as t→ 0, which is a two-dimensional analogue of radial convergence from one-dimensional dynamics. Moreover, we prove that this family of Hénon maps is stable on J and J+ when t is nonnegative.