# Eric Bedford

Professor of Mathematics
Office: Math 2-119, (631) 632-2804; Fax (631) 632-7631

E-mail: ebedford@math.stonybrook.edu

Assistants:

Dawn Huether (IMS), Math 5D-148, 632-7318

### Main Areas of Research: Complex Analysis and Complex Dynamics in higher dimension

Figures at right are examples of Fatou-Julia sets arising from semi-parabolic implosion. The map being perturbed is $f_{a,c}(x,y)=(x^2+c-ay,x)$
with $a=.1$ and $c=(a+1)^2/4$, so
$f_{a,c}$ has a semi-parabolic fixed point
with semi-parabolic basin ${\cal B}$. The white/gray
region denotes the complement ${\Bbb C}^2-{\cal B}$.
The differences in the pictures arise because
the Julia sets of $f_{a+\epsilon_j,c}$ approach different
limits for different sequences $\epsilon_j\to0$.
These pictures are explained in [BSU].

### Automorphisms of Compact Complex Surfaces

• Picture of a rational surface automorphism.

### Teaching

MAT 655 Lectures on complex Hénon maps (Fall 2016)

MAT 555 Introduction to dynamical systems (Spring 2017)

We set $f(x,y)=(y,cy+y^{-1}-\delta x )$ where
$\delta\sim .53101$ is a root of the Salem polynomial $t^5-t^4-t^3-t^2-t+1$,
and $c=2\sqrt{\delta}\cos(2\pi/5)\sim .45036$.
This induces a rational surface automorphism of a blowup of the projective plane ${\Bbb P}^2$.
The colored regions in the picture below show basins of attraction for $f$.
Does almost every point belong to a basin?