Professor of Mathematics

Institute for Math Sciences at Stony Brook

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

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].

- [BS] Series: the basic papers written with John Smillie, together with some brief introduction.
- Semi-parabolic implosion
- Real and Complex Horseshoes
- Conservative maps.
- Smoothness of the Julia set
- Expository papers

- Picture of a rational surface automorphism.

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?

My daughter Iris thinks that this design should be made into an oriental rug.