This is joint work with Danny Calegari and Sarah Koch.

In 1985, Barnsley and Harrington defined a "Mandlebrot set" $M$ for pairs of complex dilations. This is the set of complex numbers $c$ such that the limit set generated by the pair of dilations $x\mapsto cx+1$ and $x\mapsto cx-1$ is connected. The set $M$ is also the closure of the set of roots of polynomials with coefficients in $\{-1,0,1\}$. As with the usual Mandlebrot set, $M$ has strong connections to dynamics and algebra, and it has been studied by Bousch, Bandt, Solomyak, Xu, Thurston, and Tiozzo. We prove that $M$ has infinitely many "holes", that is, exotic components of Schottky space. We also prove a conjecture of Bandt that the interior of $M$ is dense in $M$.

Sarah Koch recently spoke in the seminar about this topic. I will cover similar background material (so having attended her talk is not a prerequisite) but focus on different results (so having attended her talk is not a problem). Fun pictures will be provided.