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].

Conservative (volume-preserving) maps

The Fatou set of a conservative map consists of rotation domains.
A number of interesting questons (unsolved problems) arise in the context of these rotation domains.
A description of this has been written up in the
paper Conservative maps,
which has in fact appeared as Fatou components for conservative surface automorphisms in the volume
Geometric Complex Analysis in honor of Kang-Tae Kim's 60th birthday.
(Warning: this file is large and may take a long time to download.)

Root finding algorithms

The secant method for root finding leads quite naturally to a dynamical system that is a rational surface map.
Here is an introduction to the study of this map:
Secant method

Long standing interests:

Hénon maps

[BS] Series: the basic papers written with John Smillie, together with
some brief introduction.

MAT 655/656 Lectures on complex Hénon maps (Fall 2016, Spring 2018)

MAT 555 Introduction to dynamical systems (Spring 2017, Spring 2020)

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.

...

My daughter
Sibyl
thinks that men should work more.