Speaker
Description
For $\beta\in \mathbb{C}$ with $|\beta|<1$ define the contractions
$h_0(z)=\beta z$ and $h_1(z)=\beta z+1$
and consider the attractor $A_\beta$ of the iterated function system $\{h_0,h_1\}$. In 1985 Barnsley and Harrington introduced the ``Mandelbrot set for pairs of linear maps'' which is the set of all $\beta$ with connected attractor $A_\beta$. This set has been thoroughly studied by many authors.
In the present talk we consider a variant of this Mandelbrot
set. In particular, we consider the attractors of the
iterated function system $\{f_0,f_1\}$ given by
[
f_0(z)=\beta z,\; f_1(z)=1+\beta^2z
]
and study the associated Mandelbrot set ${\mathcal M}$. Among other things we show that ${\mathcal M}$ is connected.
The structure of the iterated function system $\{f_0,f_1\}$ is related to the Fibonacci Language which is the subshift of finite type over $\{0,1\}$ defined by forbidding the occurrence of two consecutive ones. This language and its difference language play an important role in the construction.
