Raluca Tanase

Ph.D. (2013) Cornell University

First Position

Milnor Lecturer, Institute for Mathematical Sciences, Stony Brook University


Hénon maps, discrete groups and continuity of Julia sets


Research Area

complex dynamics


Consider the family of complex H\'enon maps $H(x,y) = (p(x) - ay, x)$, where $p$ is a quadratic polynomial and $a$ is a complex parameter. Let $\mathbb{D}$ be the open complex unit disk and $U^+$ the set of points that escape to infinity under forward iterations of the Hénon map. The analytic structure of the escaping set $U^+$ is well understood from previous work of Hubbard and Oberste-Vorth as a quotient of $(\mathbb{C}-\overline{\mathbb{D}}) \times \mathbb{C}$ by a discrete group of automorphisms $\Gamma$ isomorphic to $\mathbb{Z}[1/2]/\mathbb{Z}$. However, the boundary $J^+$ of $U^+$ is a complicated fractal object, on which the Hénon map behaves chaotically.

We show how to extend the group action to $\mathbb{S}^1\times\mathbb{C}$, in order to represent $J^+$ as a quotient of $\mathbb{S}^1\times \mathbb{C}/\Gamma$ by an equivalence relation. We analyze this extension for Hénon maps that are small perturbations of hyperbolic polynomials with connected Julia sets.

The proof uses results of Lyubich and Robertson about the critical locus for Hénon maps and it requires a careful analysis of the invariants of the Hénon map. We discovered an invariant function of the Hénon map, which we called $\alpha$. We developed and implemented an algorithm for computing $\alpha$. We studied the degeneracy of $\alpha$ as the Jacobian tends to $0$, and came up with an interesting relation connecting the function $\alpha$ with the group action on $\mathbb{S}^1\times\mathbb{C}$.

Hénon maps with a semi-parabolic fixed point, and their perturbations in $\mathbb{C}^2$, play an important role in understanding the parameter space of complex Hénon maps. A fixed point is semi-parabolic if the derivative of the Hénon map at that fixed point has one eigenvalue $\lambda=e^{2\pi i p/q}$ and one eigenvalue smaller than one in absolute value. Semi-parabolic implosion in $\mathbb{C}^2$ has been recently studied by Bedford, Smillie and Ueda. They show how to perturb from semi-parabolic H\'enon maps such that $J$ does not move continuously with the parameters.

Let $\lambda$ be a root of unity. For $t\geq0$ real and sufficiently small, we look at the parameter space $\mathcal{P}_{(1+t)\lambda}$ of complex Hénon maps which have a fixed point with one eigenvalue $(1+t)\lambda$. When $t=0$ these are semi-parabolic. We prove that the parametric region $\{(c,a)\in \mathcal{P}_{\lambda}: 0< |a|<\delta\}$ of semi-parabolic Hénon maps lies in the boundary of a hyperbolic component of the Hénon connectedness locus. We show that for $(c,a)\in \mathcal{P}_{(1+t)\lambda}$ and $0<|a|<\delta$, the Julia sets $J$ and $J^{+}$ depend continuously on the parameters as $t\rightarrow 0^{+}$.