Araceli Bonifant, John Milnor
On Real and Complex Cubic Curves

An expository description of smooth cubic curves in the real or complex projective plane.


T. Firsova, M. Lyubich, R. Radu, and R. Tanase
Hedgehogs for neutral dissipative germs of holomorphic diffeomorphisms of $(\mathbb{C}^{2},0)$

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 and S. Merenkov
Quasisymmetries of the basilica and the Thompson group
Abstract We give a description of the group of all quasisymmetric self-maps of the Julia set of $f(z)=z^2-1$ that have orientation preserving homeomorphic extensions to the whole plane. More precisely, we prove that this group is the uniform closure of the group generated by the Thompson group of the unit circle and an inversion. Moreover, this result is quantitative in the sense that distortions of the approximating maps are uniformly controlled by the distortion of the given map.
R. Radu and R. Tanase
Semi-parabolic tools for hyperbolic Henon maps and continuity of Julia sets in $\mathbb{C}^2$

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 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.