**Institute for Mathematical Science
Stony Brook University
**

R. Tanase, **Complex Hénon maps and discrete groups**,
Advances in Mathematics 295,
53-89 (2016).

arXiv:1503.03665.
hide/see abstract.

Consider the standard family of complex Hénon maps \(H(x,y)=(p(x)-ay,x)\),
where \(p\) is a quadratic polynomial and \(a\) is a complex parameter. Let \(U^+\) be
the set of points that escape to infinity under forward iterations.
The analytic structure of the escaping set \(U^+\) is well understood from
previous work of J. Hubbard and R. Oberste-Vorth as a quotient of \((\mathbb{C}-\overline{\mathbb{D}})\times\mathbb{C})\)
by a discrete group of automorphisms \(\Gamma\) isomorphic to \( \mathbb{Z}[1/2]/\mathbb{Z} \).
On the other hand, 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 the set \(J^+\) 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 or
polynomials with a parabolic fixed point.

R. Radu, R. Tanase, **Semi-parabolic tools for hyperbolic Hénon maps and continuity of Julia sets in $\mathbb{C}^2$**, to appear in Transactions of the AMS.
arXiv:1508.03625.
hide/see abstract.

We prove some new continuity results for the Julia sets $J$ and $J^{+}$ of
the complex Hénon map $H_{c,a}(x,y)=(x^{2}+c+ay, ax)$, where $a$ and $c$ are
complex parameters. We look at the parameter space of strongly dissipative
Hénon maps which have a fixed point with one eigenvalue $(1+t)\lambda$, where
$\lambda$ 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 $t\rightarrow 0$,
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.

R. Radu, R. Tanase, **A new proof of a theorem of Hubbard & Oberste-Vorth**,
Fixed Point Theory and Applications,

(2016) 2016:43.
arXiv:1511.03256.
hide/see abstract.

We give a new proof of a theorem of Hubbard-Oberste-Vorth for Hénon maps that are perturbations of a
hyperbolic polynomial and recover the Julia set $J^{+}$ inside a polydisk as the image of the fixed point
of a contracting operator.
We also give a different characterization of the Julia set $J^{+}$ that proves useful for later applications.

R. Radu, R. Tanase, **A structure theorem for semi-parabolic Hénon maps**, arXiv:1411.3824, Submitted.

hide/see abstract.

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 semi-parabolic fixed point with one eigenvalue $\lambda=e^{2\pi i p/q}$. We
give a characterization of those Hénon maps from the curve
$\mathcal{P}_{\lambda}$ 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 in $\mathcal{P}_{\lambda}$ 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 $\mathbb{D}_{r}\times \mathbb{D}_{r}$ it is a trivial fiber
bundle over $J_{p}$, the Julia set of the polynomial $p$, with fibers
biholomorphic to $\mathbb{D}_{r}$. The Julia set $J$ is homeomorphic to a
quotiented solenoid.

T. Firsova, M. Lyubich, R. Radu, R. Tanase, **Hedgehogs for neutral dissipative germs of holomorphic diffeomorphisms of $(\mathbb{C}^2,0)$**,
arXiv:1611.09342.
hide/see abstract.

We prove the existence of hedgehogs for germs of complex analytic
diffeomorphisms of $(\mathbb{C}^{2},0)$ with a semi-neutral fixed point at the
origin, using topological techniques. This approach also provides an
alternative proof of a theorem of Pérez-Marco on the existence of hedgehogs
for germs of univalent holomorphic maps of $(\mathbb{C},0)$ with a neutral
fixed point.

M. Lyubich, R. Radu, R. Tanase, **Hedgehogs in higher dimensions and their applications**, arXiv:1611.09840.

hide/see abstract.

In this paper we study the dynamics of germs of holomorphic diffeomorphisms
of $(\mathbb{C}^{n},0)$ with a fixed point at the origin with exactly one
neutral eigenvalue. We prove that the map on any local center manifold of $0$
is quasiconformally conjugate to a holomorphic map and use this to transport
results from one complex dimension to higher dimensions.

S. Bonnot, R. Radu, R. Tanase, **Hénon maps with biholomorphic escaping sets**, In preparation.

Last updated November 2016