**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 $\mathbb{C}^2$**,

Trans. Amer. Math. Soc. 370 (2018), 3949-3996.
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.

S. Bonnot, R. Radu, R. Tanase, **Hénon maps with biholomorphic escaping sets**,
Complex Dynamics and its Synergies (2017) 3:3,
doi.org/10.1186/s40627-017-0010-9.
hide/see abstract.

For any complex Hénon map \(H_{p,a}=(p(x)-ay,x)\), the universal cover of the forward escaping set \(U^+\) is biholomorphic to
\((\mathbb{C}-\overline{\mathbb{D}})\times\mathbb{C})\), where \(\mathbb{D}\) is the unit disk. The vertical foliation by copies of \(\mathbb{C}\)
descends to the escaping set itself and makes it a rather rigid object. In this note, we give evidence of this rigidity by showing that
the analytic structure of the escaping set essentially characterizes the Hénon map, up to some ambiguity which increases with the
degree of the polynomial \(p\).

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. Submitted.
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. Submitted.
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.

Last updated November 2017