Figures at right are examples of Fatou-Julia sets
arising from semi-parabolic implosion.
The map being perturbed is fa,c(x,y)=(x2+c−ay,x)
with a=.1 and c=(a+1)2/4, so
fa,c has a semi-parabolic fixed point
with semi-parabolic basin B. The white/gray
region denotes the complement C2−B.
The differences in the pictures arise because
the Julia sets of
fa+ϵj,c approach different
limits for different sequences ϵj→0.
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−δx) where δ∼.53101
is a root of the Salem polynomial t5−t4−t3−t2−t+1,
and c=2√δcos(2π/5)∼.45036.
This induces a
rational surface automorphism of a blowup of the projective plane P2.
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.
