Remus Radu

Ph.D. (2013) Cornell University

complex dynamics

Consider the parameter space $\mathcal{P}_{\lambda}\subset \mathbb{C}^{2}$ of complex Hénon maps

$H_{c,a}(x,y)=(x^{2}+c+ay,ax), \ \ a\neq 0$

which have a fixed point with one eigenvalue a root of unity $\lambda=e^{2\pi i p/q}$; this is a parabola in $a^{2}$. Inside the parabola $\mathcal{P}_{\lambda}$, we look at those  Hénon maps that are small perturbations of a quadratic polynomial $p$ with a parabolic fixed point of multiplier $\lambda$. We prove that there is an open disk of parameters (inside $\mathcal{P}_{\lambda}$) for which the semi-parabolic Hénon map is structurally stable on the Julia sets $J$ and $J^{+}$. The set $J^{+}$ is homeomorphic to an inductive limit of $J_{p} \times \mathbb{D}$ under an appropriate solenoidal map $\psi:J_{p} \times \mathbb{D}\rightarrow J_{p} \times \mathbb{D}$,

$\displaystyle \psi(\zeta,z)=\left(p(\zeta),\epsilon \zeta-\frac{\epsilon^{2}z}{p'(\zeta)}\right)$, where $J_{p}$ is the Julia set of the polynomial $p$. The set $J$ is homeomorphic to a solenoid with identifications, hence connected. We also consider the class of Hénon maps that are small perturbations of a hyperbolic (or parabolic) polynomial $p(x)=x^{2}+c$. We describe the set $J^{+}$ as the quotient of a 3-sphere with a dyadic solenoid removed by an equivalence relation. We define a lamination for the Hénon map by lifting the Thurston lamination of the polynomial $p$ from the closed unit disk to the unit 4-ball in $\mathbb{C}^2$, using the inductive limit. "Lifting" the leaves of the lamination of the polynomial gives a lamination for the Hénon map.